CEJM 2(4) 2004 494–508
On the Riemann zeta-function and the divisor problem Aleksandar Ivi´c∗ Katedra Matematike RGF-a, Universiteta u Beogradu, Ðuˇsina 7, 11000 Beograd, Serbia (Yugoslavia) Received 14 April 2004; accepted 19 July 2004 Abstract: Let ∆(x) denote the error term in the Dirichlet divisor problem, and E(T ) the error term in the asymptotic formula for the mean square of |ζ( 12 + it)|. If E ∗ (t) = E(t) − 2π∆∗ (t/2π) with ∆∗ (x) = −∆(x) + 2∆(2x) − 21 ∆(4x), then we obtain Z
T
0
(E ∗ (t))4 dt ¿ε T 16/9+ε .
We also show how our method of proof yields the bound R µZ X r=1
tr +G
tr −G
¶4 |ζ( 21
2
+ it)| dt
¿ε T 2+ε G−2 + RG4 T ε ,
where T 1/5+ε ≤ G ¿ T, T < t1 < · · · < tR ≤ 2T, tr+1 − tr ≥ 5G (r = 1, . . . , R − 1). c Central European Science Journals. All rights reserved. ° Keywords: Dirichlet divisor problem, Riemann zeta-function, mean square and twelfth moment of |ζ( 12 + it)|, mean fourth power of E ∗ (t) MSC (2000): 11N37, 11M06
1
Introduction and statement of results
Let, as usual, ∆(x) =
X
d(n) − x(log x + 2γ − 1) − 41 ,
n≤x ∗
E-mail:
[email protected],
[email protected]
(1)
A. Ivi´c / Central European Journal of Mathematics 2(4) 2004 494–508
and
Z
µ
T
E(T ) = 0
|ζ( 12
2
+ it)| dt − T
¶ ¡T ¢ log + 2γ − 1 , 2π
495
(2)
where d(n) is the number of divisors of n, γ = −Γ0 (1) = 0.577215 . . . is Euler’s constant. Thus ∆(x) denotes the error term in the classical Dirichlet divisor problem, and E(T ) is the error term in the mean square formula for |ζ( 21 + it)|. An interesting analogy between d(n) and |ζ( 21 + it)|2 was pointed out by F.V. Atkinson [1] more than sixty years ago. In his famous paper [2], Atkinson continued his research and established an explicit formula for E(T ) (see also the author’s monographs [7, Chapter 15] and [8, Chapter 2]). The most significant terms in this formula, up to the factor (−1)n , are similar to those in Voronoi’s formula (see [7, Chapter 3]) for ∆(x). More precisely, in [13] M. Jutila showed that E(T ) should be actually compared to 2π∆∗ (T /(2π)), where ∆∗ (x) := −∆(x) + 2∆(2x) − 12 ∆(4x). Then the arithmetic interpretation of ∆∗ (x) (see T. Meurman [16]) is X 1 (−1)n d(n) = x(log x + 2γ − 1) + ∆∗ (x). 2
(3)
(4)
n≤4x
We have the explicit, truncated formula (see e.g., [7] or [18]) √ 3 1 1 1 1X ∆(x) = √ x 4 d(n)n− 4 cos(4π nx − 14 π) + Oε (x 2 +ε N − 2 ) (2 ≤ N ¿ x). π 2 n≤N
(5)
One also has (see [7, eq. (15.68)]), for 2 ≤ N ¿ x, √ 1 3 1 1 1X ∆∗ (x) = √ x 4 (−1)n d(n)n− 4 cos(4π nx − 41 π) + Oε (x 2 +ε N − 2 ), π 2 n≤N
(6)
which is completely analogous to (5). M. Jutila, in his works [13] and [14], investigated both the local and global behaviour of E ∗ (t) := E(t) − 2π∆∗
¡ t ¢ . 2π
He proved the mean square bound Z T +H (E ∗ (t))2 dt ¿ε HT 1/3 log3 T + T 1+ε
(1 ¿ H ≤ T ),
(7)
T −H
which in particular yields Z
T
(E ∗ (t))2 dt ¿ T 4/3 log3 T.
(8)
0
Here and later ε denotes positive constants which are arbitrarily small, but are not necessarily the same at each occurrence. The bound (8) shows that, on the average, E ∗ (t)
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A. Ivi´c / Central European Journal of Mathematics 2(4) 2004 494–508
is of the order ¿ t1/6 log3/2 t, while both E(x) and ∆(x) are of the order ³ x1/4 . This follows from the mean square formulas (see e.g., [8]) Z T ∞ X 2 2 −1 ∆ (x) dx = (6π ) d2 (n)n−3/2 T 3/2 + O(T log4 T ), (9) 0
and
Z
n=1
T
2
E (x) dx = 0
2 (2π)−1/2 3
∞ X
d2 (n)n−3/2 T 3/2 + O(T log4 T ).
(10)
n=1
The mean square formulas (9) and (10) also imply that the inequalities α < 1/4 and β < 1/4 cannot hold, where α and β are, respectively, the infima of the numbers a and b for which the bounds ∆(x) ¿ xa , E(x) ¿ xb (11) hold. For upper bounds on α, β see e.g., M.N. Huxley [5]. Classical conjectures are that α = β = 1/4 holds, although this is notoriously difficult to prove. M. Jutila [13] succeeded in showing the conditional estimates: if the conjectural α = 1/4 holds, then this implies that β ≤ 3/10. Conversely, β = 1/4 implies that ∆∗ (x) ¿ε xθ+ε holds with θ ≤ 3/10. Although one expects the maximal orders of ∆(x) and ∆∗ (x) to be approximately of the same order of magnitude, this does seem difficult to establish. In what concerns the formulas involving higher moments of ∆(x) and E(t), we refer the reader to the author’s works [6], [7] and [10] and D.R. Heath-Brown [4]. In particular, note that [10] contains proofs of Z T Z T 2π 3 4 (∆∗ (t))3 dt + O(T 5/3 log3/2 T ), E (t) dt = 16π Z
0 T
Z 4
E (t) dt = 32π 0
0 T 2π
5
(∆∗ (t))4 dt + O(T 23/12 log3/2 T ).
(12)
0
In a recent work by P. Sargos and the author [12], the asymptotic formulas of K.-M. Tsang [19] for the cube and the fourth moment of ∆(x) were sharpened to Z X ∆3 (x) dx = BX 7/4 + Oε (X β+ε ) (B > 0) (13) 1
and
Z
X
∆4 (x) dx = CX 2 + Oε (X γ+ε )
(C > 0)
(14)
1 45 with β = 75 , γ = 23 . This improves on the values β = 47 , γ = 23 , obtained in [19]. 12 28 ∗ Moreover, (13) and (14) remain valid if ∆(x) is replaced by ∆ (x), since their proofs used nothing more besides (5) and the bound d(n) ¿ε nε . Hence from (12) and the analogues of (13)–(14) for ∆∗ (x), we infer then that Z T E 3 (t) dt = B1 T 7/4 + O(T 5/3 log3/2 T ) (B1 > 0), Z0 T E 4 (t) dt = C1 T 2 + Oε (T 23/12+ε ) (C1 > 0). (15) 0
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The main aim of this paper is to provide an estimate for the upper bound of the fourth moment of E ∗ (t), which is the first non-trivial upper bound for a higher moment of E ∗ (t). The result is the following Theorem 1.1. We have
Z
T
(E ∗ (t))4 dt ¿ε T 16/9+ε .
(16)
0
Note that the bounds (8) and (16) do not seem to imply each other. For the proof of (16) we shall need several lemmas, which will be given in Section 2. The proof of Theorem 1.1 will be given in Section 3. Finally, in Section 4, it will be shown how the method of proof of Theorem 1.1 can give a proof of Theorem 1.2. Let T 1/5+ε ≤ G ¿ T, T < t1 < · · · < tR ≤ 2T, tr+1 − tr ≥ 5G (r = 1, · · · , R − 1). Then R µZ X r=1
tr +G tr −G
|ζ( 21
¶4 + it)| dt ¿ε T 2+ε G−2 + RG4 T ε . 2
The bound in (17) easily gives the well-known bound (see Section 4) Z T |ζ( 12 + it)|12 dt ¿ε T 2+ε ,
(17)
(18)
0
due to D.R. Heath-Brown [3] (who had log17 T instead of the T ε -factor). It is still essentially the sharpest result concerning high moments of |ζ( 12 + it)|. General sums of zeta-integrals over short intervals, analogous to the one appearing in (17), were treated by the author in [9].
2
The necessary lemmas
Lemma 2.1. (O. Robert–P. Sargos [17]). Let k ≥ 2 be a fixed integer and δ > 0 be given. Then the number of integers n1 , n2 , n3 , n4 such that N < n1 , n2 , n3 , n4 ≤ 2N and 1/k
|n1
1/k
+ n2
1/k
− n3
1/k
− n4 | < δN 1/k
is, for any given ε > 0, ¿ε N ε (N 4 δ + N 2 ).
(19)
Lemma 2.2. Let 1 ¿ G ¿ T / log T . Then we have Z ∞ 2 2 2 E(T + u) e−u /G du + O(G log T ), E(T ) ≤ √ πG 0 and
2 E(T ) ≥ √ πG
Z
∞ 0
2 /G2
E(T − u) e−u
du + O(G log T ).
(20)
(21)
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Proof of Lemma 2.2. The proofs of (20) and (21) are analogous, so only the former will be treated in detail. From (2) we have, for 0 ≤ u ¿ T , 0≤
³ ¡ ´ ¢ +u |ζ( 12 + it)|2 dt = (T + u) log T2π + 2γ − 1 ³ ¡ ¢ ´ T −T log 2π + 2γ − 1 + E(T + u) − E(T ).
R T +u T
(22) (23)
This gives E(T ) ≤ E(T + u) + O(u log T ), hence Z
Z
G log T
−u2 /G2
E(T ) e
G log T
du ≤
0
2 /G2
(E(T + u) + O(u log T )) e−u
du.
0
The proof of (20) is completed when we extend the integration to [0, ∞) making a small R∞ R∞ √ 2 2 2 2 error, and recall that 0 e−u /G du = 12 πG, 0 ue−u /G du = 12 G. Lemma 2.3. Let 1 ¿ G ¿ T . Then we have Z ∞ ¡ ¢ ¡T 2 u ¢ −u2 /G2 ∗ T ∆ =√ ∆∗ ± e du + Oε (GT ε ). 2π 2π 2π πG 0
(24)
Proof of Lemma 2.3. Both the cases of the + and − sign in (24) are treated analogously. For example, we have ¡ ¢
¿
R
¡
¢
2 2 ∞ 1√ T u ∗ T πG∆ − 0 ∆∗ 2π + 2π e−u /G du 2 2π ³ ´ ¢ −u2 /G2 ¡T R∞ u + 2π e du = 0 ∆∗ (T ) − ∆∗ 2π ¢¢ R G log T ¡ ∗ ¡ T 2 2 T u = 0 ∆ 2π ) − ∆∗ ( 2π + 2π e−u /G du + O(1) ¯ n¯ o R G log T ¯P ¯ n (−1) d(n) + O((1 + |u|) log T ) du 2 2 ¯ ¯ T ≤n≤ π (T +u) 0 π
¿ε G2 T ε ,
where we used (4) and d(n) ¿ε nε . This establishes (24). The next lemma is F.V. Atkinson’s classical explicit formula for E(T ) (see [2], [7] or [8]). Lemma 2.4. Let 0 < A < A0 be any two fixed constants such that AT < N < A0 T , and let N 0 = N 0 (T ) = T /(2π) + N/2 − (N 2 /4 + N T /(2π))1/2 . Then E(T ) = Σ1 (T ) + Σ2 (T ) + O(log2 T ), where Σ1 (T ) = 21/2 (T /(2π))1/4
X
(−1)n d(n)n−3/4 e(T, n) cos(f (T, n)),
(25)
(26)
n≤N
Σ2 (T ) = −2
X n≤N 0
d(n)n−1/2 (log T /(2πn))−1 cos(T log T /(2πn) − T + π/4),
(27)
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499
with ¡p ¢ √ f (T, n) = 2T arsinh πn/(2T ) + 2πnT + π 2 n2 − π/4 √ √ = − 41 π + 2 2πnT + 16 2π 3 n3/2 T −1/2 +a5 n5/2 T −3/2 + a7 n7/2 T −5/2 + . . . ,
(28)
n o−1 p e(T, n) = (1 + πn/(2T ))−1/4 (2T /πn)1/2 arsinh ( πn/(2T ) = 1 + O(n/T ) √ and arsinh x = log(x + 1 + x2 ).
(1 ≤ n < T ),
Lemma 2.5. (M. Jutila [13]). For A ∈ R we have ³√ ´ Z ∞ √ √ √ 3/2 −1/2 1 3 cos 8πnT + 6 2π n T +A = α(u) cos( 8πn( T + u) + A) du,
(29)
(30)
−∞
where α(u) ¿ T 1/6 for u 6= 0, α(u) ¿ T 1/6 exp(−bT 1/4 |u|3/2 )
(31)
for u < 0, and ¡ ¢ α(u) = T 1/8 u−1/4 d exp(ibT 1/4 u3/2 ) + d¯exp(−ibT 1/4 u3/2 ) + O(T −1/8 u−7/4 )
(32)
for u ≥ T −1/6 and some constants b (> 0) and d.
3
The proof of Theorem 1.1
We shall prove that
Z
2T
(E ∗ (t))4 dt ¿ε T 16/9+ε ,
(33)
T
which easily implies (16) on replacing T by T /2, T /22 , . . . etc. and summing all the results. Henceforth we assume that T ≤ t ≤ 2T , T ε ≤ G ¿ T 5/12 , and we begin by evaluating the integrals Z ∞ 2 2 E(t ± u)e−u /G du (34) 0
which appear in Lemma 2.2 (with t replacing T ), truncating them at u = G log T with a negligible error. A similar procedure was effected by D.R. Heath-Brown [4] and by the author [7, Chapter 7], where the details of analogous estimations may be found. It transpires that the contribution of Σ2 (T ) (see (27)) in Atkinson’s formula, as well as the contribution of n in Σ1 (T ) which satisfy n > T G−2 log T will be ¿ G log T , if we take in Lemma 2.4 N = T for E(t) when T ≤ t ≤ 2T . What remains clearly corresponds to the truncated formula (6) for ∆∗ (x) with N = T G−2 log T , or equivalently r T G = log T . (35) N
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We combine now (20) with (24) with the + sign (when E(T ) ≥ 0) or (21) with (24) with the − sign (when E(T ) ≤ 0), to obtain by the Cauchy-Schwarz inequality Z ∗
2
(E (t)) ¿ε G
G log T
−1
2 /G2
e−u
(E ∗ (t + u))2 du + G2 T ε ,
(36)
−G log T
provided that T ≤ t ≤ 2T, T ε ¿ G ¿ T 5/12 . Keeping in mind the preceding discussion we thus have (replacing (t + u)1/4 with t1/4 by Taylor’s formula, with the error absorbed by the last term in (37)) by using (6), (25), (35) and (36), Z ∗
2
(E (t)) ¿ε G
−1
G log T
2 /G2
e−u
−G log T
(Σ23 (X; u) + Σ24 (X, N ; u) + Σ25 (X, N ; u)) du
+T 1+ε N −1 ,
(37)
where we set Σ3 (X; u) := t1/4 ×
X
(−1)n d(n)n−3/4
n≤X
n
· e(t + u, n) cos(f (t + u, n)) − cos( X
Σ4 (X, N ; u) := t1/4
X
Σ5 (X, N ; u) := t
(38)
(−1)n d(n)n−3/4 e(t + u, n) cos(f (t + u, n)),
X
o p 8πn(t + u) − π/4) ,
(−1)n d(n)n−3/4 cos(
p 8πn(t + u) − π/4),
(39)
X
where we suppose that (N = N (T ) is the analogue of N in (5) and (6) (cf. (36)), and not of N in Lemma 2.4) T ε ≤ X < T 1/3 , max(X, T 1/6 log T ) < N ¿ T 11/17 .
(40)
Here X = X(T ) is a parameter which allows one (by using (28)) to replace, in Σ3 (X; u), cos(f (t + u, n)) by p (1 + cn3/2 (t + u)−1/2 ) cos( 8πn(t + u) − π/4) plus terms of a lower order of magnitude. Note that, for n ≤ X (< T 1/3 ), we may also replace e(t + u, n) in (26) by 1 with the error absorbed by the last term in (37). The conditions imposed in (40) imply that G (see (36)) satisfies G ¿ T 5/12 . Hence instead of Σ3 (X; u) in (37), we may estimate X p Σ6 (X; u) := t−1/4 (−1)n d(n)n3/4 cos( 8πn(t + u) − π/4), (41) n≤X
p which has the advantage because the cosine contains 8πn(t + u) − π/4 instead of the more complicated function f (t + u, n). Thus with the aid of (37)–(41) we see that the
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left-hand side of (33) is majorized by the maximum taken over |u| ≤ G log T times Z
2T T
(E ∗ (t))2 (Σ24 (X, N ; u) + Σ25 (X, N ; u) + Σ26 (X; u) + T 1+ε N −1 ) dt ½Z
Z
2T
¿ε
∗
2T ³
4
(E (t)) dt T 7/3+ε
+T
T
Σ44 (X, N ; u)
+
Σ45 (X, N ; u)
¾1/2
´
+
Σ46 (X; u)
dt
N −1 ,
(42)
where we used the Cauchy-Schwarz inequality for integrals and (8). Thus from (42) we have the key bound Z 2T Z 2T ³ ´ 4 4 4 ∗ 4 (E (t)) dt ¿ε max Σ4 (X, N ; u) + Σ5 (X, N ; u) + Σ6 (X; u) dt |u|≤G log T
T
+T
7/3+ε
N
T −1
.
(43)
To evaluate the integrals on the right-hand side of (43) we note first that Z
2T ³ T
Σ44 (X, N ; u)
Z
´
5T /2
+ . . . dt ≤ T /2
³ ´ 4 ϕ(t) Σ4 (X, N ; u) + . . . dt,
(44)
where ϕ(t) is a smooth, nonnegative function supported in [T /2, 5T /2] , such that ϕ(t) = P P P 1 when T ≤ t ≤ 2T . The integrals of 44 (X, N ; u), 45 (X, N ; u) and 46 (X; u) are all estimated analogously. The sums over n are divided into ¿ log T subsums of the form P K
0), where √ √ √ √ √ ∆ := 8π( m + n − k − l ). (45) Therefore, in the case of Σ5 (X, N ; u), there remains the estimate Z
2T T
¯ ¯ ¯
Z Σ45 (X, N ; u) dt P∗
¿ε 1 + T
(−1)
m+n+k+l
1+ε
max
5T /2
sup
|u|≤G log T X≤K≤N
ϕ(t)× T /2
d(m)d(n)d(k)d(l)(mnkl)
−3/4
¯ √ ¯ exp(i∆ t + u)¯ dt, (46)
K<m,n,k,l≤K 0 ≤2K
P where ∗ means that |∆| ≤ T ε−1/2 holds. Now we use Lemma 2.1 (with k = 2, δ ³ K −1/2 |∆|), estimating the integral on the right-hand side of (46) trivially. We obtain that the left-hand side of (46) is ¿ε T 1+ε
max
X≤K≤N,|∆|≤T ε−1/2
K −3 T (K 4 K −1/2 |∆| + K 2 )
¿ε T ε (T 2 N 1/2 T −1/2 + T 2 X −1 ) ¿ε T 3/2+ε N 1/2 + T 2+ε X −1 .
(47)
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Proceeding analogously as in (47), we obtain that Z
2T T
Σ46 (X; u) dt ¿ε T 1+ε
max
1≤K≤X,|∆|≤T ε−1/2
T −1 K 3 (K 4 K −1/2 |∆| + K 2 )
¿ε T ε (T −1/2 X 13/2 + X 5 ),
(48)
since instead of (mnkl)−3/4 in (46) now we shall have (mnkl)3/4 t−1 (see (41)). The estimation of Σ4 (X, N ; u) (see (39)) presents a technical problem, since the cosines contain the function f (t, n), and Lemma 2.1 cannot be applied directly. First we note that, by using (28), we can expand the exponential in power series to get rid of the terms a5 n5/2 t−3/2 + . . . . In this process the main term will be 1, and the√error terms will make √ a contribution which will be (for shortness we set a = 8π, b = 16 2π 3 and τ = t + u) Z ¿ε
max
5T /2
sup T
|u|≤G log T X≤K≤N
¯ X ¯ ϕ(t)¯ (−1)n d(n)n7/4 τ −3/2 ×
T /2
³
K
× exp ia(nτ )1/2 + ib(n3 /τ ) Z ε−5 9/2 ¿ε max sup T K |u|≤G log T X≤K≤N
¯ dt
5T /2
¯ X ¯ ϕ(t)¯ (−1)n d(n)n7/4 ×
T /2
³
K
´¯2 1/2 3 1/2 ¯ × exp ia(nτ ) + ib(n /τ ) ¯ dt X ¿ε max sup T ε−5 K 9/2 (T n7/2 |u|≤G log T X≤K≤N
+T
K
X
1/2
7/4
(mn)
√ √ | m − n|−1 )
K<m6=n≤2K
¿ε
max
|u|≤G log T X≤K≤N
¿ε max T X≤K≤N
X
sup T ε−5 K 9/2 (T K 9/2 + +T 1/2 K 4
ε−5
K
9/2
TK
9/2
¿ε T
ε−4
9
N ¿ε T
|m − n|−1 )
K<m6=n≤2K 3/2+ε 1/2
N
for N ¿ T 11/17 , which is implied by (40). Thus we are left with ³√ ´ √ cos 8πnτ + 16 2π 3 n3/2 τ −1/2 − 14 π in Σ4 (X, N ; u), and we can apply Lemma 2.5. With α(v) given by (32) we have ³√ ´ √ cos 8πnτ + 16 2π 3 n3/2 τ −1/2 − A = O(T −10 ) + √ √ R u1 R∞ √ √ α(v) cos( 8πn( τ + v) − A) dv + u1 α(v) cos( 8πn( τ + v) − A) dv, (49) −u0 where we set u0 = T −1/6 log T, u1 = CKT −1/2 ,
(50)
and C > 0 is a large constant. We proceed now as in the case of Σ5 (X, N ; u). We write the cosines as exponentials in the quadruple sum over m, n, k, l. Again, after we first perform a large number of
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503
integrations by parts over t, only the portion of the sum for which |∆| ≤ T ε−1/2 will remain, where ∆ is given by (45). In the remaining sum we use (49) (once with A = 41 π and once with A = 43 π), noting that eiz = cos z + i cos(z − 12 π). We remark that, for |v| ≤ u0 , we can use the crude estimate α(u) ¿ T 1/6 , hence for this portion the estimation will be quite analogous to the preceding case. Next we note that Z u1 √ τ 1/8 v −1/4 exp(ibτ 1/4 v 3/2 ± 8πnv) dv ¿ log T (τ = t + u, |u| ≤ G log T ), u0
writing the integral as a sum of ¿ log T integrals over [U, U 0 ] with u0 ≤ U < U 0 ≤ 2U ¿ u1 , and applying the second derivative test to each of these integrals. We also remark that the contribution of the O-term in (32) will be, by trivial estimation, Z ∞ −3/4 ¿1 T −1/8 u−7/4 du ¿ T −1/8 u0 u0
if we suppose that (50) is satisfied. It remains yet to deal with the integral with v > u1 in (49), when we note that ´ ∂ ³ 1/4 3/2 √ bτ v ± 8πnv À T 1/4 v 1/2 (v > u1 ), ∂v provided that C in (50) is sufficiently large. Hence by the first derivative test Z ∞ √ √ α(v) cos( 8πn( τ + v) − 14 π) dv u1
−1/4
¿ 1 + T 1/8 u1
−1/2
T −1/4 u1
¿ 1 + T 1/4 K −3/4 ¿ 1 + T 1/4 X −3/4 ¿ 1, since K À X À T 1/3 . Thus the contribution of the integrals on the right-hand side of (49) is ¿ log T . Then we can proceed with the estimation as in the case of Σ5 (X, N ; u) to obtain Z 2T Σ44 (X, N ; u) dt ¿ε T 3/2+ε N 1/2 + T 2+ε X −1 . T
Gathering together all the bounds, we see that the integral in (33) is ³ ´ ε 3/2 1/2 2 −1 −1/2 13/2 5 7/3+ε −1 ¿ε T T N + T X + T X +X +T N ,
(51)
provided that (40) holds. Finally we choose X = T 1/3−ε ,
N = T 5/9 ,
so that (40) is fulfilled. The above terms are then ¿ε T 16/9+ε , and the proof of Theorem 1 is complete. The limit of the method is the bound ¿ T 2 X −1 ¿ T 5/3 , which would yield the exponent 5/3 + ε in (16). The true order of the integral in (16), and in general the order of the k-th moment of E ∗ (t), is elusive. This comes from the definition E ∗ (t) = E(t) − 2π∆∗ (t/(2π)), which makes it difficult to see how much the oscillations of the functions E and ∆∗ cancel each other.
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The proof of Theorem 1.2
We shall show now how the method of proof of our Theorem 1.1 may be used to yield Theorem 1.2. Our starting point is an expression for the integral Z tr +2G ϕr (t)|ζ( 21 + it)|2 dt, (52) tr −2G
where tr is as in the formulation of Theorem 1.2, and ϕr ∈ C ∞ is a non-negative function supported in [tr − 2G, tr + 2G] that equals unity in [tr − G, tr + G]. The integral in (52) majorizes the integral Z tr +G
tr −G
|ζ( 21 + it)|2 dt,
(53)
which is of great importance in zeta-function theory (see K. Matsumoto [15] for an extensive account on mean square theory involving ζ(s)). One can treat the integral in (52) by any of the following methods. (a) Using exponential averaging (or some other smoothing like ϕr above), namely the Gaussian weight exp(− 12 x2 ), in connection with the function E(T ), in view of F.V. Atkinson’s well-known explicit formula (cf. Lemma 2.4). This is the approach employed originally by D.R. Heath-Brown [3]. (b) One can use the Voronoi summation formula (e.g., see [8, Chapter 3]) for the explicit expression (approximate functional equation) for |ζ( 12 + it)|2 = χ−1 ( 12 + it)ζ 2 ( 21 + it), where ζ(s) = χ(s)ζ(1 − s), namely χ(s) = 2s π s−1 sin( 21 πs)Γ(1 − s). Voronoi’s formula is present indirectly in Atkinson’s formula, so that this approach is more direct. The effect of the smoothing function ϕr in (53) is to shorten the sum T T approximating |ζ|2 to the range 2π (1 − G−1 T ε ) ≤ n ≤ 2π (T = tr ). After this no integration is needed, and proceeding as in [7, Chapters 7-8] one obtains that the integral in (53) equals Oε (GT ε ) plus a multiple of µ ¶−1/4 Z tr +2G X 1 t k −1/2 ϕr (t) (−1) d(k)k + sin f (t, k) dt, (54) 4 2πk tr −2G 1+ε −2 k≤T
G
where f (t, k) is given by (28). (c) Instead of the Voronoi summation formula one can use the (simpler) Poisson summation formula, namely Z ∞ ∞ ∞ Z ∞ X X f (n) = f (x) dx + 2 f (x) cos(2πnx) dx, n=1
0
n=1
0
provided that f (x) is smooth and compactly supported in (0, ∞). In [11] a sketch of this approach is given. We begin now the derivation of (17), simplifying first in (54) the factor (1/4 + t/(2πk))−1/4 by Taylor’s formula, and then raising the expression in (54) to the fourth
A. Ivi´c / Central European Journal of Mathematics 2(4) 2004 494–508
505
power, using H¨older’s inequality for integrals. It follows that the sum in (17) is bounded by R trZ+2G ¯ X ¯ 4 ε −1 3 RG T + T G ϕr (t)¯
¿ε RG4 T ε + T
r=1 t −2G r Z 5T /2 −1 3
G
X
k
(−1) d(k)k
−1/4
¯4 ¯ sin f (t, k)¯ dt
k≤T 1+ε G−2
¯ ¯ ϕ(t)¯
T /2
X
¯4 ¯ (−1)k d(k)k −1/4 sin f (t, k)¯ dt,
(55)
k≤T 1+ε G−2
where ϕ(t) is a non-negative, smooth function supported in [T /2, 5T /2] such that ϕ(t) = 1 for T ≤ t ≤ 2T , hence ϕ(m) (t) ¿m T −m . Therefore it suffices to bound the expression Z 5T /2 ¯4 ¯ X ¯ k −1/4 if (t,k) ¯ (56) IK := ϕ(t)¯ (−1) d(k)k e ¯ dt, T /2
K
where T 1/3 ≤ K ¿ T 1+ε G−2 , T 1/5+ε ≤ G ≤ T 1/3 . Namely for K ≤ T 1/3 the contribution is trivially ¿ RG4 T ε , and the same holds (e.g., see [7, Theorem 7.3]) for the values G ≥ T 1/3 . Recall that √ √ f (t, k) = − 41 π + 2 2πkt + 61 2π 3 k 3/2 t−1/2 + a5 k 5/2 t−3/2 + a7 k 7/2 t−5/2 + . . . , and note that we have k 5/2 t−3/2 ¿ T 1+ε G−5 ≤ T −ε for G ≥ T 1/5+ε . This means that we may replace, on the right-hand side of (56), f (t, k) in the exponential by √ √ − 41 π + 2 2πkt + 16 2π 3 k 3/2 t−1/2 times a series whose terms are of descending order of magnitude. The main contribution will thus come from the above term. After this procedure we see that the integral in (56) bears close resemblance to the integral of the fourth moment of E ∗ (t). The term k 3/2 t−1/2 in the exponential is treated by the use of Lemma 2.5, similarly as was done in the case of Σ4 (X, N ; u) in Section 3. In our case, due to the fact that K ≥ T 1/3 may be assumed, there will be no sum corresponding to Σ3 (X; u). Now we proceed similarly as in the proof of Theorem 1.1. We shall apply Lemma 2.5 as in the proof of Theorem 1.1. Developing the fourth power in (56) and performing a large number of integrations by parts, we see that only the values for which √ √ √ √ √ |E| ≤ T ε−1/2 , E = 8π( m + n − k − l ) will be relevant, where m, n, k, l are integers from [K, K 0 ]. Thus, by Lemma 2.1 (with δ = T −1/2+ε K −1/2 ) and trivial estimation, their contribution to IK will be ¿ε T 1+ε K −1 (K 4 T −1/2 K −1/2 + K 2 ) ¿ε T 1+ε K 5/2 T −1/2 ¿ε T 3+ε G−5 . This yields the bound R µZ X r=1
tr +G
tr −G
¶4 |ζ( 12
2
+ it)| dt
¿ε RG4 T ε + G3 T −1 T 3+ε G−5 ,
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A. Ivi´c / Central European Journal of Mathematics 2(4) 2004 494–508
which is (17). It remains to show how (17) gives the twelfth moment estimate (18). Write Z 2T X |ζ( 12 + it)|12 dt ≤ |ζ( 12 + iτr∗ )|12 , T
(57)
r≤T +1
where for r = 1, 2, . . . we set |ζ( 12 + iτr∗ )| :=
max
T +r−1≤t≤T +r
|ζ( 21 + it)|.
Let {tr,V } be the subset of {τr∗ } such that V ≤ |ζ( 12 + itr,V )| ≤ 2V
(r = 1, . . . , RV ),
where clearly V may be restricted to O(log T ) values of the form 2m such that log T ≤ V ≤ T 1/6 , since ζ( 21 + it) = o(t1/6 ) (see [7, Chapter 7]). Now since we have (see e.g., [8, Theorem 1.2]), for fixed k ∈ N, Z |ζ( 12
t+ 21
k
+ it)| ¿ log t
t− 12
|ζ( 12 + it + iu)|k du + 1,
it follows that, for some points t0r (∈ [T, 2T ]) with r = 1, . . . , R0 , R0 ≤ RV , 1 ¿ G ¿ T, t0r+1 − t0r ≥ 5G, RV V
2
≤ ¿ ¿
RV X r=1 RV X r=1 R0 X
|ζ( 21 + itr,V )|2 ³Z
tr,V + 12
log T ³Z
tr,V − 12 t0r +G
log T
t0r −G
r=1
0 3/4
≤ log T (R )
¿ε T ε (RV G +
|ζ( 21
|ζ( 12
R0 Z ³X
´
2
+ it)| dt + 1 2
´
+ it)| dt + RV log T
t0r +G
|ζ( 21
2
+ it)| dt
´1/4
+ RV log T
0 r=1 tr −G 3/4 RV T 1/2 G−1/2 ),
where the estimate of Theorem 1.2 was used, with RV replacing R. If we take G = V 2 T −2ε , then we obtain 1/4 RV ¿ε T 1/2+ε G−3/2 , which gives RV ¿ε T 2+ε G−6 ¿ε T 2+ε V −12 . Then the portion of the sum in (57) for which |ζ( 12 + iτr∗ )| ≥ T 1/10+ε is ¿ log T
max
V ≥T 1/10+ε
RV V 12 ¿ε T 2+ε .
A. Ivi´c / Central European Journal of Mathematics 2(4) 2004 494–508
507
But for values of V such that V ≤ T 1/10+ε , the above bound easily follows from the large values estimate (the fourth moment) R ¿ε T 1+ε V −4 . This shows that the integral in (57) is ¿ε T 2+ε , and proves (18). Note that the author [9, Corollary 1] proved the bound R ³tX r +G X r=1
|ζ( 12 + it)|4 dt
´2
¿ RG2 log8 T + T 2 G−1 logC T
(58)
tr −G
for some C > 0, where T < t1 < . . . < tR ≤ 2T , tr+1 − tr ≥ 5G for r = 1, . . . , R − 1 and 1 ¿ G ¿ T . The bound (58), which is independent of Theorem 1.2, was proved by a method different from the one used in this work. Like (17), the bound (58) also leads to the twelfth moment estimate (18).
References [1] F.V. Atkinson: “The mean value of the zeta-function on the critical line“, Quart. J. Math. Oxford, Vol. 10, (1939), pp. 122–128. [2] F.V. Atkinson: “The mean value of the Riemann zeta-function“, Acta Math., Vol. 81, (1949), pp. 353–376. [3] D.R. Heath-Brown: “The twelfth power moment of the Riemann zeta-function“, Quart. J. Math. (Oxford), Vol. 29, (1978), pp. 443–462. [4] D.R. Heath-Brown: “The distribution of moments in the Dirichlet divisor problems“, Acta Arith., Vol. 60, (1992), pp. 389–415. [5] M.N. Huxley: Area, Lattice Points and Exponential Sums, Oxford Science Publications, Clarendon Press, Oxford, 1996. [6] A. Ivi´c: “Large values of the error term in the divisor problem“, Invent. Math., Vol. 71, (1983), pp. 513–520. [7] A. Ivi´c: The Riemann zeta-function, John Wiley & Sons, New York, 1985. [8] A. Ivi´c: The mean values of the Riemann zeta-function, LNs 82, Tata Inst. of Fundamental Research, Bombay (distr. by Springer Verlag, Berlin etc.), 1991. [9] A. Ivi´c: “Power moments of the Riemann zeta-function over short intervals“, Arch. Mat., Vol. 62, (1994), pp. 418–424. [10] A. Ivi´c: On some problems involving the mean square of |ζ( 12 +it)|, Bull. CXVI Acad. Serbe, Classe des Sciences math´ematiques, Vol. 23, (1998), pp. 71–76. ur [11] A. Ivi´c: “Sums of squares of |ζ( 21 + it)| over short intervals“, Max-Planck-Institut f¨ Mathematik, Preprint Series, Vol. 52, (2002), pp. 12. [12] A. Ivi´c and P. Sargos: On the higher moments of the error term in the divisor problem, to appear. [13] M. Jutila: “Riemann’s zeta-function and the divisor problem“, Arkiv Mat., Vol. 21, (1983), pp. 75–96; ibid. Vol. 31, (1993), pp. 61–70. [14] M. Jutila: “On a formula of Atkinson“, In: Proc. Coll. Soc. J. Bolyai–Budapest’81, Vol. 34, North-Holland, Amsterdam, 1984, pp. 807–823.
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[15] K. Matsumoto: “Recent developments in the mean square theory of the Riemann zeta and other zeta-functions“, In: Number Theory, Birkh¨auser, Basel, 2000, pp. 241–286. [16] T. Meurman: “A generalization of Atkinson’s formula to L-functions“, Acta Arith., Vol. 47, (1986), pp. 351–370. [17] O. Robert and P. Sargos: “Three-dimensional exponential sums with monomials“, J. reine angew. Math. (in print). [18] E.C. Titchmarsh: The theory of the Riemann zeta-function, 2nd Ed., University Press, Oxford, 1986. [19] K.-M. Tsang: “Higher power moments of ∆(x), E(t) and P (x)“, Proc. London Math. Soc.(3), Vol. 65, (1992), pp. 65–84.
CEJM 2(4) 2004 509–515
On the Apostol-Bernoulli Polynomials Qiu-Ming Luo∗ Department of Mathematics, Jiaozuo University, Jiaozuo City, Henan 454003, The People’s Republic of China
Received 19 July 2001; accepted 23 May 2003 Abstract: In the present paper, we obtain two new formulas of the Apostol-Bernoulli polynomials (see On the Lerch Zeta function. Pacific J. Math., 1 (1951), 161–167.), using the Gaussian hypergeometric functions and Hurwitz Zeta functions respectively, and give certain special cases and applications. c Central European Science Journals. All rights reserved. ° Keywords: Bernoulli numbers, Bernoulli polynomials, Apostol-Bernoulli numbers, ApostolBernoulli polynomials, Gaussian hypergeometric functions, Stirling numbers of the second kind, Hurwitz Zeta functions, Lerch functional equation MSC (2000): Primary: 11B68; Secondary: 33C05, 11M35, 30E20
1
Introduction
An analogue of the classical Bernoulli polynomials were defined by T. M. Apostol (see [1]) when he studied the Lipschitz-Lerch Zeta functions. We call this polynomials the Apostol-Bernoulli polynomials. First we rewrite Apostol’s definitions below Definition 1.1. Apostol-Bernoulli polynomials Bn (x, λ) are defined by means of the generating function (see [1, p.165 (3.1)] or [4, p.83]) ∞
X zexz zn = , (|z + ln λ| < 2π) B (x, λ) n λez − 1 n=0 n! setting λ = 1 in (1), Bn (x) = Bn (x, 1) are classical Bernoulli polynomials. ∗
[email protected]
(1)
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Q.-M. Luo / Central European Journal of Mathematics 2(4) 2004 509–515
Definition 1.2. Apostol-Bernoulli numbers Bn (λ) := Bn (0, λ) are defined by means of the generating function ∞
X zn z Bn (λ) , (|z + ln λ| < 2π) = λez − 1 n=0 n!
(2)
setting λ = 1 in (2), Bn = Bn (1) are classical Bernoulli nunbers. T.M.Apostol not only gave elementary properties of polynomials Bn (x, λ) in [1], but also obtained the recursion formula for the set of numbers Bn (λ) using the Stirling numbers of the second kind (see [1, p. 166 (3.7)]) as follows Bn (λ) = n
n−1 X
(−1)k k!λk (λ − 1)−1−k S(n − 1, k), (n ∈ N0 ; <(λ) > 0, λ 6= 1)
(3)
k=1
where S(n, k) denote the Stirling numbers of the second kind which are defined by means of the following expansion (see [3, p.207, Theorem B]) n µ ¶ X x x = k!S(n, k). k k=0 n
(4)
By applying binomial series expansion and Leibniz’s rule, we first obtain the representation of the polynomials Bn (x, λ) involving the Gaussian hypergeometric functions, and thereout deduce Apostol’s formula (3); afterward we prove Theorem 3.1 using Lerch functional equation with related Hurwitz Zeta function. Furthermore we show that the main result in [9, p.1529, Theorem A] is only a special case of Theorem 3.1.
2
Apostol-Bernoulli Polynomials and Gaussian Hypergeometric functions
Theorem 2.1. If n is a positive integer and <(λ) > 0, λ 6= 1 are complex numbers, then we have Bn (x, λ) = n
¶ n−1 µ X n−1 l=0
l
l
λ (λ − 1)
−l−1
l X
µ ¶ l l (−1) j (x + j)n−l−1 j j=0 j
× F [l − n + 1, l; l + 1; j/(x + j)]
(5)
where F [a, b; c; z] denotes Gaussian hypergeometric functions defined by (cf. [5, p.44 (4)]) F [a, b; c; z] :=
∞ X (a)n (b)n z n n=0
where (λ)0 = 1,
(c)n
n!
(λ)n = λ(λ + 1) · · · (λ + n − 1) =
,
|z| < 1
Γ(λ+n) , (n Γ(λ)
≥ 1).
(6)
Q.-M. Luo / Central European Journal of Mathematics 2(4) 2004 509–515
511
Proof 2.2. We differentiate both sides of (1) with respect to the variable z. Applying Leibniz’s rule yields ½ ¾¯ ¯ zexz n ¯ , Dz = d . Bn (x, λ) =Dz z λe − 1 ¯z=0 dz ½· µ ¶ ¸−1 ¾¯ (7) n X n ¯ λ n−k k−1 z −1 ¯ . kx Dz (e − 1) + 1 =(λ − 1) ¯ λ−1 k z=0 k=0
Since binomial series expansion
(1 + w)
−1
=
∞ X
(−w)l , |w| < 1
(8)
¶l k−1 µ X λ Dzk−1 {(ez − 1)l }|z=0 . 1 − λ l=0
(9)
l=0
setting w =
λ (ez λ−1
− 1), we have
Bn (x, λ) = (λ − 1)
−1
n µ ¶ X n k=0
k
kx
n−k
By the definition of Stirling numbers of the second kind (see [5, p.58 (15)]) z
l
(e − 1) = l!
∞ X
S(r, l)
r=l
zr r!
(10)
yields k−1 n µ ¶ X X n k Bn (x, λ) = (−1)l λl (λ − 1)−l−1 l!S(k − 1, l)xn−k . k l=0 k=1
(11)
We change sum order of k and l, and using the formula below (see [5, p.58 (20)]) µ ¶ k 1 X k−j k (−1) S(n, k) = jn k! j=0 j
(12)
we obtain Bn (x, λ) = n
n−1 X
l
λ (λ − 1)
−l−1 n−k−l−1
x
l=0
¶µ ¶k µ µ ¶ n−l−1 j n−1 l l X . (13) j (−1) n−k−2 x j j=0 k=0
l X
j
Applying (6) to (13) readily yields Bn (x, λ) = n
¶ n−1 µ X n−1
l=0 l X
l
λl (λ − 1)−l−1 xn−k−l−1
µ ¶ l l × (−1) j .F [l − n + 1, 1; l + 1; −j/x] j j=0 j
(14)
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Q.-M. Luo / Central European Journal of Mathematics 2(4) 2004 509–515
Finally, we apply the known transformation [10, 15.3.4] F [a, b; c; z] = (1 − z)−a F [a, c − b; c; z/(z − 1)], and (2.2) immediately obtain (5). Remark 2.3. H .M. Srivastava and P. G Todorov considered earlier the formula of the generalized Bernoulli polynomials (see [6, p.510 (3)]), for α = 1, which is a complementarity of our result (5), for λ = 1, as follows µ ¶ µ ¶ k Pn n k! X j k (−1) j 2k (x + j)n−k Bn (x) = k=0 j k (2k)! j=0 × F [k − n, k − 1; 2k + 1; j/(x + j)].
(15)
Remark 2.4. We will also apply the representation (5) in order to derive an interesting special case considered by T. M. Aspotol in (3). By the well-known formula [10, 15.1.20] F [a, b; c; 1] =
Γ(c)Γ(c − a − b) Γ(c − a)Γ(c − b)
(c 6= 0, −1, −2, . . . , <(c − a − b) > 0),
upon setting a = l − n + 1, b = l, and c = l + 1 yields µ ¶−1 n−1 F [l − n + 1, l; l + 1; 1] = , l
(0 ≤ l ≤ n).
(16)
In view of (16), the special case of our formula (5) when x = 0 gives Apostol’s representation (3). Remark 2.5. The formula of classical Bernoulli numbers considered by H. W. Gould in [7, p.49, Eq.(17)] is also a complementarity of the Apostol’s formula (3), for λ = 1 µ ¶µ ¶−1 n X n+k k n+1 Bn = (−1) S(n + k, k). (17) n − k k k=0 Remark 2.6. There is a relationship between Apostol-Bernoulli polynomials B n (x, λ) and Stirling numbers of the second kind: µ ¶X n k−1 X n Bn (x, λ) = k (−1)j λj (λ − 1)−j−1 j!S(k − 1, j)xn−k , (λ 6= −1). (18) k j=0 k=1 Remark 2.7. Recently, Luo also obtained the relation between the classical Bernoulli polynomials and the Stirling numbers of the second kind [8], which is a complementarity of (18), for λ = 1: ¶µ ¶µ ¶−1 n X n−k µ X n k + s + 1 s + 2k (19) Bn (x) = S(s + 2k, k)xn−s−k . s + k s k k=0 s=0
Q.-M. Luo / Central European Journal of Mathematics 2(4) 2004 509–515
3
513
Apostol-Bernoulli polynomials and the Hurwitz Zeta Function
It is well-known that Hurwitz-Lerch Zeta functions are defined by the infinite series (see [5, p.121, (1)]) Φ(z, a, s) =
∞ X k=0
(a ∈ C \
Z− 0 ;s
zk , (a + k)s
(20)
∈ C, when |z| < 1; <(s) > 1, when |z| = 1).
Setting z = e2πiz gives the Lipschitz-Lerch Zeta function φ(z, a, s) (see [1, p.161]) ∞ X e2kπiz φ(z, a, s) = , s (a + k) k=0
(21)
(a ∈ C \ Z− 0 ; z ∈ R \ Z, <(s) > 0; z ∈ Z, <(s) > 1). When s = −n is negative integer, the Lipschitz-Lerch Zeta function φ(z, a, s) was evaluated by T. M. Apostol using polynomials Bn (x, λ) (see [1, p.164]): φ(z, a, −n) = −
Bn+1 (a, e2πiz ) . n+1
(22)
If we set z = 0 in (22), then we have the formula (see [2, p.264, Theorem 12.13]) ζ(a, −n) = −
Bn+1 (a) n+1
where ζ(a, s) are Hurwitz Zeta functions defined by ζ(a, s) :=
(23) P∞
k=0
1 (see [2, (a + k)s
p.249]) and Bn (x) are classical Bernoulli polynomials. Further, if we set a = 0 in (23), then we have the known formula (see [2, p.266, Theorem 12.16]): Bn+1 (24) ζ(−n) = − n+1 P 1 where ζ(s) are Riemann Zeta functions defined by ζ(s) := ∞ k=0 s (see [2, p.249]) and k Bn := Bn (0) are classical Bernoulli numbers. In this section, we will apply the Lerch functional equation to obtain the representap tion of Apostol-Bernoulli polynomials Bn (x, λ) at rational points x = . Clearly, it is a q generalized form of [9, p.1529, Theorem A]. Theorem 3.1. For n ∈ N \ {1}; p ∈ Z, q ∈ N, z ∈ C we have ( q ´ h³ n 2(z + j − 1)p ´ i ³p X ³z + j − 1 ´ n! 2πiz πi Bn , e =− , n exp − ζ q (2qπ)n j=1 q 2 q ) q h³ n 2(j − z)p ´ i ³j − z ´ X πi . , n exp − + (25) + ζ q 2 q j=1
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Q.-M. Luo / Central European Journal of Mathematics 2(4) 2004 509–515
Proof 3.2. In view of the Lipschitz-Lerch Zeta functions φ(z, a, s) (21), and using the known series identity ∞ X k=1
f (k) =
q ∞ X X
f (qk + j),
(q ∈ N)
(26)
j=1 k=0
we obtain that φ(z, a, s) = q
−s
µ ¶ q X a+j−1 ζ , s exp(2(kq + j − 1)πiz), q j=1
p yields q µ µ ¶ ¶ µ ¶ q X a+j−1 p 2(j − 1)pπi −s ζ φ , a, s = q , s exp , q q q j=1
(k ∈ N0 ).
(27)
(p ∈ Z, q ∈ N).
(28)
Setting z =
On the other hand, by the Lerch functional equation (see [5, p.125, (29)]), for 0 < z < 1, 0 < a < 1, s ∈ C ´i h ³1 Γ(s) n exp πi s − 2az φ(−a, z, s) φ(z, a, 1 − s) = (2π)s 2 h ³ 1 ´i o + exp πi − s + 2a(1 − z) φ(a, 1 − z, s) . (29) 2 p Setting a = above and applying (28) yields q ( q h³ s 2(z + j − 1)p ´ i Γ(s) X ³ z + j − 1 ´ p , s exp − ζ πi φ(z, , 1 − s) = q (2qπ)s j=1 q 2 q ) q h³ s 2(j − z)p ´ i ³j − z ´ X πi , (p ∈ Z, q ∈ N).(30) + , s exp − + ζ q 2 q j=1 Finally, set s = n in (30), and Apostol’s formula (22) leads immediately to formula (25). Remark 3.3. If z in Theorem 3.1 is an integer, then we deduce readily the Cvijovic and Klinowski’s result (see [9, p.1529, Theorem A]) Bn
³p´ q
q ³ 2jpπ nπ ´ 2 · n! X ³ j ´ , n cos − =− ζ , (2qπ)n j=1 q q 2
(31)
(n ∈ N \ {1}; p ∈ N0 , q ∈ N; 0 ≤ p ≤ q).
Acknowledgements The author appreciate the anonymous referee and the editor, Professor Monika Sperling, for their valuable comments and addition to references. The author was supported in part by NNSF (#10001016) of China, SF for the Prominent Youth of Henan Province China (#0112000200).
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References [1] T.M. Apostol: “On the Lerch Zeta function”, Pacific J. Math., Vol. 1, (1951), pp. 161–167. [2] T.M. Apostol: Introduction to analytic number theory, Springer-Verlag, New York/Heidelberg/Berlin, 1976. [3] L. Comtet: Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel, Dordrecht/Boston, 1974. (Translated from the French by J.W. Nienhuys) [4] H.M. Srivastava: “ Some formulae for the Bernoulli and Euler polynomials at rational arguments”, Math. Proc. Cambridge Philos. Soc., Vol. 129, (2000), pp. 77–84. [5] H.M. Srivastava and Junesang Choi: Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht/Boston/London, 2001. [6] H.M. Srivastava, P.G. Todorov: “An explicit formula for the generalized Bernoulli polynomials”, J. Math. Anal. Appl., Vol. 130, (1988), pp. 509–513. [7] H.W. Gould: “Explicit formulas for Bernoulli numbers” Amer. Math. Monthly, Vol. 79, (1972), pp. 44–51. [8] Qiu-Ming Luo: “The Bernoulli Polynomials Involving the Gaussian Hypergeometric Functions”, [submitted]. [9] D. Cvijovic and J. Klinowski: “New formula for The Bernoulli and Euler polynomials at rational arguments”, Proc. Amer. Math. Soc., Vol. 123, (1995), pp. 1527–1535. [10] M. Abramowitz and I.A. Stegun (Eds): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 4th printing, Washington, 1965.
CEJM 2(4) 2004 516–526
Distinguished geodesics and Jacobi fields on first order jet spaces Vladimir Balan∗ , Nicoleta Voicu† 1
Department Mathematics I, University Politehnica of Bucharest, Splaiul Independent¸ei 313, RO-060042 Bucharest, Romania 2 University ”Transilvania” of Bra¸sov, Str. Iuliu Maniu nr. 50, RO-500091 Bra¸sov, Romania
Received 4 April 2004; accepted 7 July 2004 Abstract: In the framework of jet spaces endowed with a non-linear connection, the special curves of these spaces (h-paths, v-paths, stationary curves and geodesics) which extend the corresponding notions from Riemannian geometry are characterized. The main geometric objects and the paths are described and, in the case when the vertical metric is independent of fiber coordinates, the first two variations of energy and the extended Jacobi field equations are derived. c Central European Science Journals. All rights reserved. ° Keywords: jet space, nonlinear connection, Cartan connection, energy, geodesic, stationary curve, path, Jacobi field MSC (2000): 58A20, 53C22, 53B15
The geometrized framework on first and higher-order osculating spaces was introduced and widely studied by Acad. R.Miron and collaborators ([13]). As a complementary extension of the tangent (first-order osculating) framework, in the last decade, there was developed with significant results the geometric approach on first-order jet spaces ([18], [17], [3]).
1
Basic objects of the geometrized jet framework
Let ξ = (E = J 1 (T, M ), π, T × M ) be the first order jet bundle of mappings ϕ : T → M , where T and M are C ∞ real differentiable manifolds (dim T = m, dim M = n). The local ∗ †
[email protected] [email protected], [email protected]
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jet coordinates on E will be denoted by (tα , xi , y A )(α,i,A)∈I∗ ≡ (y µ )µ∈I , where the set of indices I splits as follows I = Ih ∪ Iv , Ih = Ih1 ∪ Ih2 , Iv = m + n + 1, m + n + mn Ih1 = 1, m,
Ih2 = m + 1, m + n, I∗ = Ih1 × Ih2 × Iv .
and the indices implicitly take values as described below: α, β, · · · ∈ Ih1 ; i, j, · · · ∈ Ih2 ; A, B, · · · ∈ Iv ; λ, µ, · · · ∈ I. As well, when appropriate, we identify A = m + n + n(i − m − 1) + α as A ≡ ( iα ) and i ∂xi denote y A ≡ x(α ) = ∂t α. We endow E with a non-linear connection N = {NµA }µ∈Ih ,A∈Iv which determines the local adapted basis of X (E), B = {δα , δi , δA }(α,i,A)∈I∗ ≡ {δµ }µ∈I , with ∂α = ∂t∂α , ∂i = ∂x∂ i and δα = ∂α − NαA δA ,
δi = ∂i − NiA δA ,
δ A = ∂A =
∂ . ∂y A
(1)
The dual basis of B writes then B ∗ = {δ α , δ i , δ A }(α,i,A)∈I∗ ≡ {δ µ }µ∈I , where δ α = dtα , δ i = dxi , δ A ≡ δy A = dy A + NαA dtα + NiA dxi .
(2)
Any d-linear connection ([4, 6, 17]) ∇ = {Lλµν }λ,µ,ν∈I on E has its components relative to the adapted basis provided by the relations δ λ (∇δν δµ ) = Lλµν , ∀λ, µ, ν ∈ I. The coefficients of a linear connection are ∇ ≡ {Lλµν } = {Lαβγ , Lαβk , LαβC , Lijγ , Lijk , LijC , LABγ , LABk , LABC }. Among these connections which preserve the two hrizontal and vertical submodules of sections in X (E), one finds in the presence of a metrical structure on E the Cartan linear ik connection, which is metrical and satisfies the conditions ([18], [17]) Lijγ = g2 ∂γ gjk , Li[jk] = Li ³k] ´ = 0. We shall further consider the case when hαβ (t) and gij (t, x) are two [j
α
non-degenerate N -tensor fields of constant signature on T and M respectively, and hence we may endow E with a semi-Riemannian metric G = hαβ (t)dtα ⊗ dtβ + gij (t, x, y)dxi ⊗ dxj + g˜AB (t, x, y)δy A ⊗ δy B , {z } {z } | {z } | | g˜
g
h
where g˜AB ≡ g˜(iα )(j ) = hαβ (t)gij (t, x, y). β The Cartan connection on (E, G) has then the coefficients c
λ
α
i
i
i
A
A
A
∇≡ {L µν } = {L βγ , 0, 0, Ljγ , Ljk , LjC , L Bγ , L Bk , L BC },
(3)
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which are given by ¯ α ¯ 1 αε ¯ = h (δ{β hε}γ − δε hβγ ), Li = 1 g il (δ{k gj}l − δl gjk ), Lαβγ = ¯βγ jk 2 2 Lijγ = 12 g ik δγ gkj , Lij
= 1 g il (δ³{k ´ gj}l − δ(lγ ) gjk ), γ (kγ ) 2 i i ¯β ¯ ¯ ¯ (iα ) ¯ , L(αj ) = δαβ ¯ i ¯ , L(αj ) = δαβ Li , Lα = Lα = 0. L j γ = δαβ Lijγ − δji ¯αγ jC jk βj βC (β ) (β ) k (β )C
(4)
c
The adapted components of the torsion T and of the curvature R of ∇ are defined by the relations λ δ λ (T (δν , δµ )) = Tµν , δ λ (R(δν , δµ )δρ ) = Rρ λµν , ∀ λ, µ, ν, ρ ∈ I.
Then the Cartan essential torsion coefficients are, for the case of g dependent on x only ([17], [18, Theorem 4.4]) (iα ) (iα ) (iα ) i {Tγ j , Tk j , T j k , Tβ i j , TjA , Tβ Aγ , Tβ Aj , Ti Aj }. (β ) ( β ) (γ ) (β ) The five essential and three derived nontrivial sets of curvature N -tensor fields are respectively {Rβαγδ , Rj i km , Rj i γλ , Rj i λA , Rj i CD },
{RB Aγδ , RB Aλk , RB AµC },
for λ ∈ Ih , µ ∈ I. We shall investigate especially the ARLS (almost Riemannian Lagrange separated) case where the coefficients gij depend only on x and g is a Riemannian metric on M ; in ∗
this case the Cartan connection ∇ has just four nontrivial sets of coefficients ¯α ¯ ¯i ¯ ¯ ¯ (iα ) ¯ ¯ ∗ (iα ) λ α i i β β i ∇≡ {L µν } = {Lβγ = ¯βγ ¯ , 0, 0, 0, Ljk = ¯jk ¯}, 0, L(j )γ = −δj ¯αγ ¯ , L(j )k = δα ¯jk ¯ , 0} β
β
and we have (see the diagram below; [17])
(kγ ) i i A T βk = −Liβk = 0, T jC = LijC = 0, T BC = T(i ) j = δγα C ki(j ) − δγβ C kj(iα ) = 0. α (β ) β c
∗
c
∇ := ∇ for h∗ (t) ⊗ g(x)
∇ for h∗ (t) ⊗ g(t, x, y) hT
hM
v
hT
hM
v
hT hT
0
0
A T βγ
hT hT
0
0
A T βγ
hM hT
0
i T βk
A T βk
hM hT
0
0
A T βk
hM hM
0
0
A T jk
hM hM
0
0
A T jk
vhT
0
0
A T βC
vhT
0
0
A T βC
vhM
0
i T jC
A T jC
vhM
0
0
A T jC
vv
0
0
A T BC
vv
0
0
0
Table 1 The torsions of the Cartan connection.
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2
519
Paths and stationary curves on J 1 (T, M )
Consider on J GLn = (E, g˜) a fixed nonlinear (Cartan-Ehresmann) connection N , and let ∇ be a linear d−connection on E; we endow E with the metric G induced by two non-degenerate d−tensor fields h ∈ T20 (T ) and g ∈ T20 (M ). Let c : J = [a, b] ⊂ R → E be a smooth curve, whose image lies in a chart U˜ ⊂ E, c(s) = (tα (s), xi (s), y A (s)) ≡ (y µ (s)), ∀t ∈ J. Definition 2.1. µ a) The field V = δy δµ defined on c will be called d−velocity field of the curve c. The ds components of V are explicitely given by µ
{V }µ∈I ≡
µ
δy A = y˙ A + NβA t˙β + NjA x˙ j t , x˙ , ds ˙α
i
¶
, (α,i,A)∈I∗
where we denote by dot the s-derivation. We have also denoted by A = Aµ δµ , where Aµ =
∇V µ def δV µ + Lµνρ V ν V ρ , = ds ds
the d−acceleration on c, which provides the motion of the test-body along c. b) We shall say that c is a stationary curve with respect to ∇ iff A = 0 along the curve. c) The curve c is called h−curve, if πv (V) = 0, and v−curve, if πh (V) = 0, where by πh and πv we denoted respectively the h− and v−projectors of the canonic splitting induced by N . If a h − /v−curve satisfies also the extra condition A = 0, then it is called h − /v−path, respectively. Analytically, these curves are described by the following Theorem 2.2. (Balan [4]) Let c : J ⊂ R → E be a curve. Then the following hold true: a) c is an h−curve iff V A = 0. The h−curve is an h−path iff it satisfies dV µ + Lµνρ V ν V ρ = 0, ∀µ ∈ Ih . ds
(5)
b) c is a v−curve iff V µ = 0, ∀µ ∈ Ih . The v−curve is a v−path iff δV A B C + LA BC V V = 0, ∀A ∈ Iv . ds
(6)
It should be mentioned that the implicit sum in the right term of (5) and (6) involves just horizontal/vertical index types, respectively. The particular uniparametric autonomous case of J GLn provides the known corresponding paths from the tangent framework for Finslerian, Lagrange and Generalized Lagrange structures (see e.g. [14, 3, 9, 21]).
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The first variation of energy. Geodesics in J GLn
We consider now the general case, and define the geodesics of J GLn as the C ∞ extremals of the energy E. Let G be a Riemannian metric on E given as in (3), N a nonlinear c
connection and let ∇ =∇ be the associated Cartan connection on (E, N, G). To find the equations of geodesics, we consider a piecewise regular curve c• : J = [a, b] ⊂ R → E, smooth on the intervals Ir = [sr , sr+1 ], r = 0, k − 1, where a = s0 < s1 < · · · < sk = b. Consider as well a variation of c• which is piecewise smooth on Ir , r = 0, k − 1 , given by c : Iε = (−ε, ε) × [a, b], c(0, s) = c• (s) ≡ (cµ (s)µ∈I , ∀s ∈ [a, b], with fixed ends: c(u, a) = c• (a), c(u, b) = c• (b), ∀u ∈ Iε . Denote ω(u) = c(u, · ) : ∂c ∂c , cu = ∂u , [a, b] → E, ∀u ∈ Iε and cs = ∂s dcµ δcµ δcµ dc• = • ∂µ = • δµ ; W = cu |u=0 = |u=0 δµ , ds ds ds du and let h · , · i be the metric bilinear form Gc• (s) ( · , · ). The energy of the curve ω(u), is given by Z V = cs |u=0 = c˙• =
b
E(u) =
hcs , cs ids, u ∈ Iε .
a
Then we have the following Theorem 3.1. ([7, 22, 23]) The first variation of the energy is given by Z b k−1 X 1 dE(u) (hT (W, V), Vi − hW, ∇V Vi)ds, hW, ∆r Vi + |u=0 = − 2 du a r=1
(7)
where ∆r V = lim V(s) − lim V(s) and T is the torsion of ∇. s&sr
s%sr
Proof. Denote ∂s =
∂ , ∂s
∂u =
∂ , ∂u
∇s =
∇ , ∂s
∇u =
∇ ∂u
and ∆r cs = lim cs (u, s) − s&sr
lim cs (u, s). Since ∇ is metrical, we have
s%sr
∂s hcs , cu i = h∇s cs , cu i + hcs , ∇s cu i,
∂u hcs , cs i = 2h∇u cs , cs i.
As well, from ∂˙su = ∂˙us we get [cu , cs ] = 0, and hence ∇u cs = T (cu , cs ) + ∇s cu and ∇W V = T (W, V) + ∇V W. Then Rb Rb Rb dE(u) d = du hc , c ids = 2 a h∇u cs , cs ids = 2 a hT (cu , cs ) + ∇s cu , cs ids a s s du Rb Rb = 2 a hT (cu , cs ), cs )ids + 2 a (∂s hcu , cs i − hcu , ∇s cs i)ds =2
Rb a
hT (cu , cs ), cs )ids + 2
= −2hcu , cs i|
k−1 X
hcu , cs i
|ssrr+1
r=0
|ba
−2
k−1 X r=1
hcu , ∆r cs i + 2
Z
−2
Z
b
hcu , ∇s cs ids a
b
(hT (cs , cu ), cs i − hcu , ∇s cs i)ds, a
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where the scalar product is induced by the metric G at c(u, s). Then, for u = 0 replacing cs |u=0 = V, cu |u=0 = W, and using that W(a) = W(b) = 0, we get the relation (7). Remark 3.2. 1. It is known ([12, 11]) that in the case when the metrical connection ∇ is torsionless, then the condition E(0) = 0 satisfied for all the variations of c• as above, implies that c• is a geodesic of the metric space (i.e., minimizer of both the energy and length functionals, see [12]); in this case the geodesics are shown to be smooth curves, satisfying the condition ∇V V = 0, where V = c˙• .
(8)
Hence a natural extension in the jet framework is to define as stationary curves (or dgeodesics, [7]) of J GLn the smooth autoparallel curves of ∇, i.e., which obey (8); these are the autoparallel curves of the Cartan connection. The second name is justified, since in the autonomous case for m = 1 these project onto (proper) geodesics of M , provided that they are h−paths (Anastasiei and Buc˘ataru [1]). 2. Still, considering the field F defined by the equation hT (W, V), Vi = hW, Fi, the first variation becomes Z b k−1 X 1 dE(u) |u=0 = − hW, F − ∇V Vids, hW, ∆r Vi + 2 du a r=1
(9)
and hence the proper geodesics of J GLn are the smooth curves which satisfy the equations ([22, 23]) ∇V λ = F, with F µ = g µρ gλτ V ν V τ T νρ , (10) ds where gλτ ∈ {hαβ , gij , g˜AB }. We have γ λ k C F α = hβα gλτ V ν V τ T νβ = hβα {hγδ V ν V δ T νβ + gkl V ν V l T νβ + g˜CD V ν V D T νβ } γ C k λ } + g˜CD V ν V D T νj = g ji {hγδ V ν V δ T νj + gkl V ν V l T νj F i = g ji gλτ V ν V τ T νj γ λ k C F A = geBA gλτ V ν V τ T νB = geBA {hγδ V ν V δ T νB + gkl V ν V l T νB + g˜CD V ν V D T νB }. ∗
We note that in the particular case of the Cartan connection ∇, the only remainig nonzero C . terms of the torsion are just T µρ
4
Special geodesics
We subsequently consider the special curves of the J 1 (T, M )-framework and denote with c
∇ the Cartan connection ∇. I. hT -geodesics (”temporal geodesics”), xi = xi0 (=constant). Using V i = V A = 0,
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these are shown to satisfy: dV α γ + Lαβγ V β V γ = hβα hγδ V ε V δ T εβ ds F i = 0 ⇔ hγδ V ε V δ T εjγ = 0 A
ε
F = 0 ⇔ hγδ V V
δ
γ T εB
= 0,
(11) (12) (13)
a system of m + n + mn equations with m + mn unknown components. In particular, for ∗ γ γ = T εjγ = T εB = 0, i.e., the restrictions (12)-(13) the Cartan connection ∇, we have T εβ are identically satisfied, and the equations (11) rewrite as dV α + Lαβγ V β V γ = 0. ds This proves that the following statements are equivalent: 1) c is an hT -geodesic; ∗
2) c is an hT -autoparallel of ∇. ¯α ¯ ¯ and consequently the Moreover, if h is a Riemannian metric on T , then Lαβγ = ¯βγ statements 1) and 2) are equivalent also with 3) c is an hT -curve which projects to a geodesic of T . II. For hM -geodesics (”spatial geodesics”), using V α = V A = 0 (⇒ tα = tα0 - constant), we infer DV i = F i , F A = 0, F α = 0, ds which rewrite k k gkl V h V l T hβ = 0, gkl V h V l T hB =0 i dV k + Lijk V j V k = g ji gkl V h V l T hj . ds ∗
(14) (15)
k k k = T hj = T hB = 0, and hence, the restrictions (14) Example 4.1. For ∇, we have T hβ n are identically satisfied by any curve on J GL ; this shows that for an hM -curve, the following are equivalent: 1) c is a geodesic; ∗
2) c is an autoparallel curve of ∇; 3) c projects to a geodesic of the Riemannian manifold M . III. For h-geodesics, we have V A = 0, whence dV α ds
+ Lαβγ V β V γ + Lαβk V β V k = F α
dV i ds
+ Lijγ V j V γ + Lijk V j V k = F i , F A = 0.
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∗
Example 4.2. For ∇, the equations above (considering V A = 0) lead to dV α ds
+ Lαβγ V β V γ + Lαβk V β V k = 0
dV i ds
+ Lijγ V j V γ + Lijk V j V k = 0, F A = 0.
C Since F A = g˜AE g˜CD V σ V D TσE , it follows that for horizontal curves the condition F A = 0 is identically satisfied, i.e., c is a horizontal geodesic if c is a horizontal autoparallel curve ∗ of ∇.
IV. The v-geodesics: (V α = V i = 0 ⇒ tα , xi - constant) satisfy the system with the unknown components y A F α = 0, F i = 0,
δV A + LABC V B V C = F A , ds
which lead to C C g˜CD V E V D TEβ = 0, geCD V E V D TEj =0 A δV C + LABC V B V C = g˜BA g˜CD V E V D TEB . ds
(16) (17)
∗
C = 0, and hence the v-geodesics (in case these exist) are those v-paths For ∇, we have TEB ∗
of ∇ which obey the conditions (16).
5
The second variation of the energy. Jacobi fields
In the study of geodesics, an important tool for locating conjugate points along geodesics and describing geodesic variations are the Jacobi fields. We define an alternative to [7] analogous notion for Jacobi fields in the d−framework on J GLn , emerging from the second variation of the energy functional (integral of the square of arc-length, [12, 14]). Consider J GLn endowed with a nonlinear connection and the Cartan connection ∇. Let c• be a d−geodesic and c : Iε × Iε × [a, b] a piecewise variation with two parameters of c• , satisfying similar conditions as the variation c in the Theorem above. Denote Wi = cui |(u1 ,u2 )=(0,0) , i = 1, 2. Then we have the following Theorem 5.1. The second variation of the energy (the Hessian) is given by k−1 X 1 ∂ 2 E(u1 , u2 ) hW2 , ∆r (T (W1 , V) + ∇V W1 )i |(u1 ,u2 )=(0,0) = − 2 ∂u1 ∂u2 r=1 Rb + a hW2 , ∇W1 F − ∇2V W1 − R(W1 , V)V − ∇V T (W1 , V)ids.
(18)
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V. Balan and N. Voicu / Central European Journal of Mathematics 2(4) 2004 516–526 ∂c , ∂ui
Proof. We denote cui = 1 ∂E(u1 , u2 ) = 2 ∂u2
Z
b a
i = 1, 2 and u = (u1 , u2 ). Then
Z b k−1 X hcu2 , ∇s cs ids, hT (cu2 , cs ), cs ids − hcu2 , ∆r cs i − a
r=1
which rewrite, for hT (W, V)Vi = hW, Fi 1 ∂E(u1 , u2 ) = 2 ∂u2
Z
b
hcu2 , F − ∇s cs ids − a
k−1 X
hcu2 , ∆r cs i,
r=1
and hence k−1 X 1 ∂ 2 E(u1 , u2 ) (h∇u1 cu2 , ∆r cs i + hcu2 , ∇u1 ∆r cs i)+ =− 2 ∂u1 ∂u2 r=1
+
Rb a
(h∇u1 cu2 , F − ∇s cs i + hcu2 , ∇u1 F − ∇u1 ∇s cs i)ds,
where F˜ = F˜ µ δµ is the vector field given by the relation hT (cu2 , cs ), cs i = hF, cu2 i, or locally, λ ν τ cs cs |(u1 ,u2 ,s) . F˜ µ = g µρ gλτ T νρ ˜ 0, s) = F(s) and ∆r cs |u=(0,0) = ∆r V, r = 1, k − 1, since c• is C 1 on Then we have F(0, [a, b]. As well, c• being a d−geodesic implies ∇s cs |u=(0,0) = ∇V V = F, and we obtain Z b k−1 X 1 ∂ 2 E(u1 , u2 ) |u=(0,0) = − hW2 , ∆r ∇W1 Vi + hW2 , ∇W1 F − ∇W1 ∇V Vids. 2 ∂u1 ∂u2 a r=1
(19)
Since ∇W1 V = T (W1 , V) + ∇V W1 and ∇W1 ∇V V = R(W1 , V)V + ∇V (T (W1 , V) + ∇V W1 ) + ∇[W1 ,V] V, {z } | ∇W 1 V
where the last term cancels on c• , the last term in (19) becomes Z b hW2 , ∇W1 F − ∇2V W1 − R(W1 , V)V − ∇V T (W1 , V)ids, a
which plugged in (19) leads to (18). The theorem suggests the following natural generalization of the concept of Jacobi field for the d−framework. Definition 5.2. A d−vector field J on E is called d−Jacobi field if it satisfies the equation ∇J F = ∇2V J + R(J, V)V + ∇V T (J, V). Locally, this equation rewrites ∇2 J µ ∇T µ + = Rµ + J λ ∇δλ F µ , µ ∈ Ih ∪ Iv , ds2 ds
(20)
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µ where we denote T µ = V λ J σ T λσ , Rµ = −V ρ V λ J σ Rµρλσ .
Remarks. For ∇J F = 0, the autonomous J GLn case for m = 1 leads in particular to the extended concept of Jacobi field proposed in the GLn framework by Anastasiei and Buc˘ataru ([2]). As well, we note that in the Riemannian case, the h − part of a d−Jacobi field coincides with the classical one ([12, 10, 8, 20]). Let B(W1 , W2 ) = h∇V W1 , W2 i − hW1 , ∇V W2 i; then B(W1 , W2 ) = −B(W2 , W1 ), ∀W1 , W2 ∈ X (E). Hence being skew-symmetric, B defines a pre-symplectic structure on the set of Jacobi fields along the geodesics of E, as in the classical case ([10, 20]).
Acknowledgment The present work was partially supported by Grant CNCSIS MEN 33552 (75) / 1.07.2003.
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[11] S.Kobayashi, K.Nomizu: Foundations of Differential Geometry I, II, Interscience Publishers, New York, 1963, 1969. [12] J. Milnor: Morse Theory, Ann. of Math. Stud., Princeton Univ. Press, 1963. [13] R. Miron: The Geometry of Lagrange Spaces: Theory and Applications, Kluwer Acad. Publishers, 1994. [14] R. Miron, M. Anastasiei: The Geometry of Vector Bundles. Theory and Applications, Kluwer, Dordrecht, 1994. [15] R. Miron, M. Tatoiu-Radivoiovici: “A Lagrangian theory of electromagnetism”, Rep. Math. Phys., Vol. 27, (1989), pp. 49–84. [16] M. Neagu: “The geometry of autonomous metrical multi-time Lagrange space of electrodynamics”, International Journal of Mathematics and Mathematical Sciences, Hindawi Publishing Corporation, (2001), http://xxx.lanl.gov/abs/math.DG/0010091, (2000). [17] M. Neagu: ”Generalized metrical multi-time Lagrangian geometry of physical fields“, http://xxx.lanl.gov/abs/math.DG/0011003, (2000). [18] M. Neagu, C. Udri¸ste: “The geometry of metrical multi-time Lagrange spaces”, http://xxx.lanl.gov/abs/math.DG/0009071, (2000). [19] D.J. Saunders: The Geometry of Jet Bundles, Cambridge University Press, 1989. [20] Z. Shen: Differental Geometry of Sprays and Finsler Spaces, Kluwer Acad. Publishers, 2001. [21] P.C. Stavrinos, H. Kawaguchi: “Deviation of geodesics in the gravitational field of Finslerian Space-Time”,Memoirs of Shonan Inst. of Technol., Vol. 27, (1993), pp. 35–40. [22] N. Voicu: “On metrical linear connections with torsion in Riemannian geometry”, An.S¸t.Univ. ”Al.I.Cuza”, Ia¸si, [submitted]. [23] N. Voicu: “The Exponential Map on the Second Order Tangent Bundle”, Studia Mathematica, University of Cluj, [submitted].
CEJM 2(4) 2004 527–537
A compound of the generalized negative binomial distribution with the generalized beta distribution Tadeusz Gerstenkorn∗ University of Trade, Pojezierska St 97b, 91-341 L Ã ´od´z, Poland
Received 15 September 2003; accepted 2 September 2004 Abstract: This paper presents a compound of the generalized negative binomial distribution with the generalized beta distribution. In the introductory part of the paper, we provide a chronological overview of recent developments in the compounding of distributions, including the Polish results. Then, in addition to presenting the probability function of the compound generalized negative binomial–generalized beta distribution, we present special cases as well as factorial and crude moments of some compound distributions. c Central European Science Journals. All rights reserved. ° Keywords: compound distributions, generalized negative binomial distribution, generalized beta distribution, factorial and crude moments. MSC (2000): 60
1
Introduction
The problem of the compounding of probability distributions dates back to the 1920s. It is worth noting the papers by M. Greenwood and G.U. Yule (1920; the parameter of the Poisson distribution treated as a gamma variable) and by E.S. Pearson (1925; a device for determining the parameter of a distribution in Bayes’ theorem). The problem was addressed in the 1940s by L. Lundberg (1940; the P´olya distribution treated as a compound distribution), by F.E. Satterthwaite (see a remark in W. Feller’s (1943, p. 390) paper on the comparison of Satterthwaite’s idea of the generalized Poisson distribution with a compound distribution), by W. Feller (1943; some compound distributions treated as “contagious” distributions), by G. Skellam (1948; the binomial–beta distribution furnish∗
E-mail: [email protected], [email protected]
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T. Gerstenkorn / Central European Journal of Mathematics 2(4) 2004 527–537
ing the P´olya–Eggenberger distribution), and by M.E. Cansado (1948; compound Poisson distributions). In the 1950s, J. Gurland (1957) explored some interrelationships among compound and generalized distributions. Important theoretical problems were considered in the 1960s by H. Teicher (1960, 1962), by W. Molenaar (1965), and by W. Molenaar ´ odka (1964a; and W.R. van Zwet (1966). Also in the 1960s we find the works of T. Sr´ Rayleigh with gamma and gamma with Rayleigh; 1966; Laplace with normal N(0, 3) left truncated at zero, gamma, Rayleigh, Weibull, and exponential), of S.K. Katti and J. Gurland (1967; Poisson–Pascal), and of D.S. Dubey (1968; compound Weibull). Then, from the 1970s onward, as the 20th Century waned, we have papers by D.S. Dubey (1970; ´ odka (1972; generalized gamma compound gamma, beta, and F), T. Gerstenkorn and Sr´ with gamma), by H. Jakuszenkow (1973; Poisson with normal N(m, σ) left truncated at zero, generalized two-parameter gamma, Maxwell and Rayleigh), by W. Dyczka (1973; binomial with beta (incomplete moments)), by T. Gerstenkorn (1982; binomial with generalized beta; 1993; generalized gamma with exponential; 1996; P´olya with beta), by G.E. Wilmot (1989; Poisson–Pascal and others), and by M.L. Huang and K.Y. Fung (1993; D compound Poisson distribution as a new extension of the Neyman Type A distribution). A broad range of relevant references can be found in the review studies by N.L. Johnson, S. Kotz, and A. Kemp (1992), by G.P. Patil et al. (1968), and most recently by G. Wimmer and G. Altmann (1999).
2
The compounding of probability distributions
We begin by giving a definition and the relations needed for the compounding of distributions. Definition 2.1. Let Xy be a random variable with a distribution function F (x|y) that depends on parameter y. Suppose parameter y is a random variable Y with distribution function G(y). Then the distribution that has the distribution function of X defined by the formula Z∞ F (x|cy)dG(y) H(x) = (1) −∞
will be called compound, where c is an arbitrary constant or a constant bounded on some interval (J. Gurland (1957)). The occurrence of the constant c in (1) has a practical justification inasmuch as the distribution of a random variable, in describing a phenomenon, often depends on a parameter that is itself a realization of another random variable multiplied by a certain constant. The variable that has distribution function (1) will be symbolized by ”X ∧ Y ” and will be called a compound of the variable X with respect to the “compounding” Y .
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Relation (1) will be symbolized as follows: H(x) ≡ F (x|cy) ∧ G(y).
(2)
Y
Consider the case when one variable is discrete, with probability function P (X = xi |cy) where parameter y is a random variable Y with density g(y). Then (1) is expressed by the formula Z∞ g(y)P (X = xi |cy)dy. h(xi ) = P (X = xi ) = (3) −∞
3
The compounding of the generalized negative binomial distribution with the generalized beta distribution
3.1 The generalized negative binomial ∧ the generalized beta Y
G.C. Jain and P.C. Consul (1971) define the generalized negative binomial distribution (GNBD). W. Dyczka (1978) introduces a correction to the assumptions given by Jain and Consul and also deals with the GNBD as a distribution of the power series (PSD) class. Other assumptions for the GNBD (not needed here) can be found in G. Wimmer and G. Altmann (1999, p. 297). Definition 3.1. The generalized negative binomial distribution is a distribution with the probability function given by the formula n n + βx x n+βx−x , x = 0, 1, 2, . . . , GN B(x; n, p, β) = Pβ (x, n, p) = p (1 − p) n + βx x (4) where 0 ≤ p < 1,
n > 0,
0 ≤ p ≤ 1,
βp < 1,
n ∈ N,
β ≥ 1,
β = 0.
(i) (ii)
For β = 1 one obtains from (4) the negative binomial distribution. If n ∈ N , for β = 0 one obtains from (4) the binomial distribution, and for β = 1, the Pascal distribution (in accordance with N. Johnson, S. Kotz, and A. Kemp (1992, p. 200)). Definition 3.2. By the generalized beta distribution (GBD) we mean a distribution given by the density function ¡ ¢ ay r−1 y a w−1 1 − bw for 0 < y < (bw)1/a , (bw)r/a B(r/a,w) GB(y; a, b, w, r) = (5) 1/a 0 for y ≤ 0 or y ≥ (bw)
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T. Gerstenkorn / Central European Journal of Mathematics 2(4) 2004 527–537
where a, b, r, w > 0 and B(r/a, w) is a beta function. Distribution (5) is a special limit case of the Bessel distribution investigated by T. ´ Sr´odka (1973). It was also analyzed by J. Seweryn (1986) and by W. Ogi´ nski (1979) and was applied in reliability theory. Let us consider the GNBD (4) that depends on cy: n n + βx x n+βx−x Pβ (x; n, cy) = , x = 0, 1, 2, . . . , (6) (cy) (1 − cy) n + βx x where 0 < cy < 1, n = 1, 2, . . . , βcy < 1, β ≥ 1, and Y is a random variable with GBD (5). Theorem 3.3. The probability function of the compound distribution GNBD ∧ GBD Y
is given by the formula ∞ X
k n + βx − x (−c) Pβ GB(x) = D (bw) a B k k=0
where
D= x = 0, 1, 2, . . . ;
µ
k
n n + βx
¶ x+r+k ,w , a
(7)
x n + βx x c (bw) a x
B(r/a, w)
a, b, w, r > 0, where βcy < 1,
,
0 < cy < 1,
n > 0,
β ≥ 1.
Proof. From formulas (3), (5) and (6) we have 1/a (bw) Z
µ ¶w−1 ya Pβ GB(x) = aD1 y 1− (1 − cy)n+βx−x dy bw 0 1/a (bw) ¶w−1 µ Z ∞ X n + βx − x ya k x+r+k−1 dy, = aD1 (−c) y 1− bw k k=0 0 x+r−1
where
n + βx x c x
D1 =
n . n + βx (bw)r/a B(r/a, w)
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Substituting y a /bw = t, we get Z1 ∞ X n + βx − x x+r+k k −1 k Pβ GB(x) = D (−c) (1 − t)w−1 dt, (bw) a t a k k=0 0 x = 0, 1, 2, . . . ,
a, b, w, r > 0,
n > 0,
β ≥ 1,
0
1 β(bw)1/a
and, from the definition of the beta function, we obtain (7). In the special case when n, β ∈ Nwe have a simpler formula given by µ ¶ n+βx−x X n + βx − x k x+r+k k Pβ GB(x) = D (−c) , w . (8) (bw) a B a k k=0 Case (ii) of (4) – the case of β = 0 – is discussed next in Section 3.2.
3.2 Special cases In the case when β = 0 (the binomial distribution) the proof of Theorem 3.3 is similar to, but even simpler than, the proof given above. More specifically, we get µ ¶ n−x X n − x k x+r+k k ∗ (−c) Pβ GB(x) = D , w (bw) a B a k k=0 where
D∗ =
x n x c (bw) a x B(r/a, w)
.
Here we get a compound of the binomial distribution with the generalized beta distribution: binomial ∧ generalized beta (β = 0) Y x n x c (bw) a µ ¶ n−x x X n − x k x+r+k k P0 GB(x) = ,w , (9) (−c) (bw) a B B(r/a, w) k=0 a k x = 0, 1, 2, . . . , n. T. Gerstenkorn (1982, p. 90, (6)) obtained the same result. Moreover, Gerstenkorn determined some special cases of this compound–compounds of the binomial distribution with the beta distribution, generalized gamma, exponential, arcsine, Erlang, χ2 , gamma,
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Rayleigh, Maxwell, χ, normal truncated, and Weibull. In the Gerstenkorn paper, factorial moments and ordinary moments of order l of the distributions are also determined. From (8) we also have the compound of the generalized negative binomial with the beta distribution GNBD ∧ beta (b = 1/w, a = 1) Y n + βx ∞ x k X nc n + βx − x Pβ B(x) = (−1)k B(x + r + k, w). n + βx B(r, w) k=0 k It can be demonstrated that ∞ X n + βx − x (−1)k B(x + r + k, w) = B(x + r, w + n + βx − x) k k=0 holds. Hence, we get
n c n + βx B(x + r, w + n + βx − x) , n + βx B(r, w) x k
Pβ B(x) =
a result that, for β = 1 and c =1, yields negative binomial ∧ beta Y n + x − 1 B(x + r, w + n) P1 B(x) = B(r, w) x
(10)
(11)
or, after using the formula n[x,−1] = n(n + 1)(n + 2) . . . (n + x − 1), P1 B(x) =
n[x,−1] r[x,−1] w[n,−1] . x!(r + w)[x+n,−1]
Furthermore, a consequence of Theorem 3.3 is for β = 1, n = 1, 2, . . . the compound distribution Pascal binomial ∧ generalized beta Y
µ ¶ n k x+r+k n + x − 1 cx (bw) k P1 GB(x) = ,w . (−c) (bw) a B B(r/a, w) a x k k=0 x a
n X
(12)
Some special compounds, as mentioned before for binomial–generalized beta, can be obtained for distribution (12).
T. Gerstenkorn / Central European Journal of Mathematics 2(4) 2004 527–537
4
533
Factorial moments and ordinary (crude) moments of a compound of the negative binomial distribution with the generalized beta distribution
Definition 4.1. Let Xy and X be random variables with distribution functions F (x|y) and H(x), respectively (see (1)), and let parameter y have distribution G(y). Then, when one keeps in mind the formula for the so-called factorial polynomial x[l] = x(x − 1)(x − 2) . . . (x − (l − 1)), Z∞ [l] m[l] = E(X ) = E(Xy[l] )dG(y)
(13)
∞
is called a factorial moment of order l of the variable X with compound distribution (1). Relation (13) will be symbolized as follows: E(Xy[l] ) ∧ G(y).
(14)
Y
Theorem 4.2. The factorial moment of order l of the compound distribution negative binomial ∧ generalized beta is given by the formula Y µ ¶ l ∞ k l+r+k n[l,−1] cl (bw) a X −l N B−GB k m[l] ,w (15) = (−c) (bw) a B B(r/a, w) k=0 a k Proof. The factorial moment of order l of the negative binomial distribution is given by the formula µ ¶l p NB m[l] (n, p) = n[l,−1] , 0 < p < 1, q = 1 − p q (W. Dyczka (1973, p. 223, (63))). Consequently, by (14), the factorial moment of this order of distribution (12) is, if we let p = cy, the following: B−GB B mN = mN [l] (n, cy) ∧ GB(y; a, b, r, w) [l] Y
1/a (bw) Z µ
¶l µ ¶w−1 cy ya r−1 y dy 1− 1 − cy bw 0 1/a (bw) µ ¶w−1 Z ∞ X −l ya an[l,−1] cl k l+r+k−1 y 1− dy. = (−c) (bw)r/a B(r/a, w) k=0 bw k 0 [l,−1]
an = r/a (bw) B(r/a, w)
Substituting y a /bw = t, we get [l,−1] l
m[l] =
n
c (bw) B(r/a, w)
l a
∞ X
k −l k (−c) (bw) a k k=0
Z1 t 0
l+r+k −1 a
(1 − t)w−1 dt,
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T. Gerstenkorn / Central European Journal of Mathematics 2(4) 2004 527–537
a result that yields (15). A special case of Theorem 4.2 is the following theorem. Theorem 4.3. The factorial moment of order l of the compound distribution negative binomial ∧ beta (b = 1/w, a = 1, c = 1) is given by the formula (for w> l) Y
B−B mN = n[l,−1] [l]
B(l + r, w − l) B(r, w)
(16)
or by the formula B−B mN = [l]
n[l,−1] r[l,−1] . (w − 1)[l]
(17)
Proof. In this case we have B−B mN = [l]
n[l,−1] B(r, w)
∞ X
−l k (−1) B(l + r + k, w). k k=0
It can be demonstrated that ∞ X l k (−1) B(l + r + k, w) = B(l + r, w − l), k k=0 thus giving formula (16). Formula (17) is then evident. Further special cases can easily be obtained by taking account of the remarks included in Section 3. The ordinary (crude) moments of the compound distributions under consideration are obtained by using the formula l X ml = Skl m[k] , k=0
where Skl stands for the so-called Stirling numbers of the second kind. Bohlmann (1913) seems to be the first to give this formula; the tables for these numbers can be found, for instance, in A. Kaufmann (1968) or in J. L Ã ukasiewicz and M. Warmus (1956).
References [1] G. Bohlmann: “Formulierung und Begr¨ undung zweier Hilfss¨atze der mathematischen Statistik”, Math. Ann., Vol. 74, (1913), pp. 341–409. [2] M.E. Cansado: “On the compound and generalized Poisson distributions”, Ann. Math. Stat., Vol. 19, (1948), pp. 414–416.
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[3] D.S. Dubey: “A compound Weibull distribution”, Naval Res. Logist. Quart, Vol. 15(2), (1968), pp. 179–182. [4] D.S. Dubey: “Compound gamma, beta and F distributions”, Metrika, Vol. 16(1), (1970), pp. 27–31. [5] W. Dyczka: “Zastosowanie skÃladania rozkÃlad´ow do wyznaczania moment´ow (Application of compounding of distribution to determination of moments; in Polish)”, Zeszyty Nauk. Politech. L ô odzkiej – Scient. Bull. L ô od´z Techn. Univ., Vol. 168, (1973); Matematyka, Vol. 3, (1973), pp. 205–230. [6] W. Dyczka: “A generalized negative binomial distribution as a distribution of PSD class”, Zeszyty Nauk. Politech. L ô odzkiej – Scient. Bull. L ô od´z Techn. Univ., Vol. 272, (1978); Matematyka, Vol. 10, (1978), pp. 5–14. [7] W. Feller: “On general class of “contagious” distributions”, Ann. Math. Stat., Vol. 14, (1943), pp. 389–400. ´ odka: Kombinatoryka i Rachunek Prawdopodobie´ [8] T. Gerstenkorn and T. Sr´ nstwa (Combinatorics and Probability Theory; in Polish), 7th Ed., PWN, Warsaw, 1983. [9] T. Gerstenkorn: “The compounding of the binomial and generalized beta distributions”, In: W. Grossmann, G.Ch. Pflug and W. Wertz (Eds.): Proc. 2nd Pannonian Conf. on Mathem. Statistics (Tatzmannsdorf, Austria, 1981), D. Reidel Pub., Dordrecht, Holland, 1982, pp. 87–99. [10] T. Gerstenkorn: “A compound of the generalized gamma distribution with the exponential one”, Bull. Soc. Sci. Letters Lodz, Vol. 43(1), (1993); Recherches sur les d´eformations, Vol. 16(1), (1993), pp. 5–10. [11] T. Gerstenkorn: “A compound of the P´olya distribution with the beta one”, Random Oper. and Stoch. Equ., Vol. 4(2), (1996), pp. 103–110. [12] I. Ginzel: “Die konforme Abbildung durch die Gammafunktion”, Acta Mathematica, Vol. 56, (1931), pp. 273–353. [13] M. Greenwood and G.U. Yule: “An inquiry into the nature of frequency distribution representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents”, J. Roy. Stat. Soc., Vol. 83, (1920), pp. 255–279. [14] J. Gurland: “Some interrelations among compound and generalized distributions”, Biometrika, Vol. 44, (1957), pp. 265–268. [15] M.L. Huang and K.Y. Fung: “D compound Poisson distribution”, Statistical PapersStatistische Hefte, Vol. 34, (1993), pp. 319–338. [16] G.C. Jain and P.C. Consul: “A generalized negative binomial distribution”, SIAM J. Appl. Math, Vol. 21(4), (1971), pp. 507–513. [17] H. Jakuszenkow: “Nowe zÃlo˙zenia rozkÃlad´ow (New compounds of distributions; in Polish)”, Przeglad , Statystyczny, Vol. 20(1), (1973), pp. 67–73. [18] N.L. Johnson, S. Kotz and A. Kemp: Univariate Discrete Distributions, 2nd Ed., J. Wiley, New York, 1992. [19] S.K. Katti and J. Gurland: “The Poisson-Pacal distribution”, Biometrics, Vol. 17, (1967), pp. 527–538.
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[20] A. Kaufmann: Introduction `a la Combinatorique en Vue des Applications, Dunod, Paris, 1968. [21] O. Lundberg: On Random Processes and Their Application to Sickness and Accident Statistics, Almquist and Wiksells Boktryckeri–A. B., Uppsala, 1940. [22] J. L à ukaszewicz and M. Warmus: Metody Numeryczne i Graficzne (Numerical and Graphical Methods; in Polish), PWN, Warszawa, 1956. [23] W. Molenaar: “Some remarks on mixtures of distributions”, Bull. Intern. Statist. Institute; 35th Session of the Intern. Statist. Inst., Beograd, Vol. 2, (1965), pp. 764– 765. [24] W. Molenaar and W.R. van Zwet: “On mixtures of distributions”, Ann. Math. Stat., Vol. 37(1), (1966), pp. 281–283. [25] L. Ogi´ nski: “Zastosowanie pewnego rozkÃladu typu Bessela w teorii niezawodno´sci (Application of a distribution of the Bessel type in the reliability theory; in Polish)”, Zeszyty Nauk. Politech. L ô odzkiej – Scient. Bull. L ô od´z Techn. Univ., Vol. 324, (1979); Matematyka, Vol. 12, (1979), pp. 31–42. [26] G.P. Patil, S.W. Joshi and C.R. Rao: A Dictionary and Bibliography of Discrete Distributions, Oliver and Boyd, Edinburgh, 1968. [27] E.S. Pearson: “Bayes’s theorem, examined in the light of experimental sampling”, Biometrika, Vol. 17, (1925), pp. 388–442. [28] F.E. Satterthwaite: “Generalized Poisson distribution”, Ann. Math. Stat., Vol. 134, (1942), pp. 410–417. [29] J.G. Seweryn: “Some probabilistic properties of Bessel distributions”, Zeszyty Nauk. Politechn. L ô odzkiej – Scient. Bull. L ô od´z Techn. Univ., Vol. 466, (1986); Matematyka, Vol. 19, (1986), pp. 69–87. [30] J.G. Skellam: “A probability distribution derived from the binomial distribution by regarding the probability of success as variable between the sets of trials”, J. Roy. Statist. Soc., Ser. B, Vol. 10(2), (1948), pp. 257–261. [31] E.W. Stacy: “A generalization of the gamma distribution”, Ann. Math. Stat., Vol. 33(3), (1962), pp. 1187–1192. ´ odka: “Nouvelles compositions de certaines distributions”, Bull. Soc. Sci. Lettres, [32] T. Sr´ L ô od´z, Vol. 15(5), (1964), pp. 1–11. ´ odka: “ZÃlo˙zenie rozkÃladu Laplace’a z pewnym uog´olnionym rozkÃladem gamma, [33] T. Sr´ Maxwella i Weibulla (Compounding of the Laplace distribution with a generalized gamma, Maxwell and Weibull distributions; in Polish)”, Zeszyty Nauk Politech. L ô odzk.- Scient. Bull., L ô od´z Techn. Univ., Vol. 77, (1966); WÃl´ okiennictwo (Textile industry), Vol. 14, (1966), pp. 21–28. ´ odka: “On some generalized Bessel-type probability distribution”, Zeszyty Nauk. [34] T. Sr´ Politechn. L ô odzk. – Scient. Bull. L ô od´z. Techn. Univ., Vol. 179, (1973); Matematyka, Vol. 4, (1973), pp. 5–31. [35] H. Teicher: “On the mixtures of distributions”, Ann. Math. Stat., Vol. 31, (1960), pp. 55–72. [36] H. Teicher: “Identifiability of mixtures”, Ann. Math. Stat., Vol. 32, (1962), pp. 244– 248..
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[37] G.E. Willmot: “A remark on the Poisson-Pascal and some other contagious distributions”, Statist.Probab. Lett., Vol. 7(3), (1989), pp. 217–220. [38] G. Wimmer and G. Altmann: Thesaurus of Discrete Probability Distributions, Stamm, Essen, 1999.
CEJM 2(4) 2004 538–560
Zero-dimensional subschemes of ruled varieties∗ Edoardo Ballico1† , Cristiano Bocci2‡ , Claudio Fontanari1§ 1
2
Department of Mathematics, University of Trento, 38050 Povo (TN), Italy Department of Mathematics, University of Milano, I-20133 Milano, Italy
Received 22 September 2003; accepted 23 August 2004 Abstract: Here we study zero-dimensional subschemes of ruled varieties, mainly Hirzebruch surfaces and rational normal scrolls, by applying the Horace method and the Terracini method. c Central European Science Journals. All rights reserved. ° Keywords: Ruled varieties, Hirzebruch surfaces, fat points, Horace lemma, Grassmann defective varieties MSC (2000): 14N05
1
Introduction
Let Z ⊆ Pr be a closed subscheme. For any integer t, let ρZ,t,r be the restriction map H 0 (Pr , OPr (t)) → H 0 (Z, OZ (t)). We say that Z has maximal rank if for every integer t > 0 the linear map ρZ,t,r has maximal rank, i.e. it is either injective or surjective. We are going to investigate this property for a zero-dimensional Z. A very classical motivation for such an interest is provided by the following Definition 1.1. Let X ⊂ Pr be an integral nondegenerate projective variety of dimension n. The h-secant variety Sech (X) of X is the Zariski closure of the set {p ∈ Pr : p lies in ∗
This research is part of the T.A.S.C.A. project of I.N.d.A.M., supported by P.A.T. (Trento) and M.I.U.R. (Italy). † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected]
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the span of h + 1 independent points of X}. The expected dimension of Sech (X) is expdim Sech (X) = min{(n + 1)(h + 1) − 1, r} and X is h-defective with h-defect δh (X) if δh (X) = expdim Sech (X) − dim Sech (X) > 0. A basic tool for understanding defective varieties is the classical Lemma of Terracini (see [33] for the original version and [18] or [1] for a modern proof), which says that, for P1 , . . . , Ph+1 general points and P general in their span, one has TP (Sech (X)) ⊆< TP1 (X), . . . , TPh+1 (X) > with equality holding in characteristic zero. Therefore, at least over the field C, X is not h-defective if and only if the restriction map ρZ,1,r : H 0 (Pr , OPr (1)) −→ H 0 (Z, OZ (1)) has maximal rank, where Z := ∪h+1 i=1 2Pi ⊂ X. More generally, for Z := ∪h+1 i=1 mi Pi ⊂ X with mi ≥ 2, the restriction map ρZ,1,r fails to have maximal rank if and only if the union of the (mi − 1)-osculating space at Pi for i = 1, . . . , n spans a linear space of dimension less than the expected one. In other words, the linear system of hyperplane sections having a contact of order mi at Pi has dimension greater than the expected one, i.e. it is a special linear system. Here we focus on ruled varieties, by applying as an essential tool the so-called Horace method. We recall its set-up in Section 2, while in Section 3 we collect the main results of the present paper. In particular, Horace method provides a quick proof of the following well-known Proposition 1.2. Fix integers x > 0, a, b ≥ 3. Let Z ⊂ F0 be a general union of x double points of F0 . Then the restriction map ρZ,a,b : H 0 (F0 , OF0 (ah + bF )) → H 0 (Z, OZ (ah + bF )) has maximal rank. We stress that the above result is sharp: indeed, it is easy to verify that homogeneous special systems arise when we consider the value a = 2. This fact is true not only for F0 , but for Fe with e ≥ 0. For example, consider the system L := |2h + (2d + P 2Pi | on Fe , e ≥ 0; we have h0 (Fe , OFe (2h + (2d + 2e)F ) = 6d + 3e + 3, 2e)F − 2d+e+1 i=1 hence the virtual dimension of L is −1 then we would expect that L is empty. Since h0 (Fe , OFe (h+(d+e)F ) = 2d+e+2, there is a unique curve C in the system |h+(d+e)F | passing through Pi , i = 1, . . . , 2d + e + 1: its double 2C is a divisor on Fe of type 2h + (2d + 2e)F with 2d + e + 1 double points, hence 2C ∈ L and L is special. We also remark that, at least in characteristic zero, Proposition 1.2 may be proved by using [5], Lemma 4.2, or [5], Proposition 4.1, [16], Lemma 3, [32], Lemma 2.2, and
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certain reducible curves with negative intersection with KFe . From this point of view the cases e = 0, 1 are quite easier, because in this case KF∗e is ample. However, Horace method allows us to handle also fat points of higher multiplicity. For instance, we obtain the following Theorem 1.3. Fix integers b ≥ 4, a ≥ 11, s > 0 and mi , 1 ≤ i ≤ s, such that 1 ≤ mi ≤ 3 for every i. Fix s general points Pi ∈ F0 , 1 ≤ i ≤ s, and set Z := ∪mi Pi . Then the restriction map ρZ,a,b : H 0 (F0 , OF0 (ah + bF )) → H 0 (Z, OZ (ah + bF )) has maximal rank. We do not have a unified way to handle all cases with 3 ≤ a ≤ 10 left open by Theorem 1.3, hence we omit the few cases with low a which can be easily checked by hand. Indeed, the paper [26] by Antonio Laface contains the complete list of all homogeneous special systems with multiplicity two or three. We point out that such a list gives some information also in the non-homogeneous case, just by applying the Horace method (see Proposition 3.6). For Fe , e > 0, the proof of Theorem 1.3 gives a similar, but weaker, statement (Theorem 3.11); roughly speaking, we have to replace b with b − ea in our assumptions. Our work suggests several open questions, which are collected in Section 4. In particular, Question 4.4 should be viewed as the analogue for Hirzebruch surfaces of the celebrated Segre and Harbourne-Hirschowitz conjectures for the projective plane. Next, in Section 5 we consider higher dimensional scrolls and we apply the same methods to the study of a zero-dimensional subscheme which is curvilinear, i.e. its Zariski tangent space at a reduced point has dimension at most one. Finally, in Section 6 we turn to another geometrically meaningful kind of defectivity, the so-called Grassmann defectivity, which is defined as follows. Definition 1.4. Let X ⊂ Pr be an integral nondegenerate projective variety of dimension n defined over the field C. The (k, h)-Grassmann secant variety Seck,h (X) of X is the Zariski closure of the set {l ∈ G(k, r) : l lies in the span of h + 1 independent points of X}. The expected dimension of Seck,h (X) is expdim Seck,h (X) = min{(h + 1)n + (k + 1)(h − k), (k + 1)(r − k)} and X is (k, h)-defective with (k, h)-defect δk,h (X) if δk,h (X) = expdim Seck,h (X) − dim Seck,h (X) > 0. This property was classically investigated by Alessandro Terracini in his beautiful paper [34], going back to 1915. Recent advances in the subject are due to Ciro Ciliberto, Luca Chiantini, and Marc Coppens ([10], [11], [12], and [17]); see also the papers [19], [21], and [20], all three strongly inspired by the pioneering work of Terracini. Here we give a new application of the same method in order to answer the following Question 1.5. (Ciliberto, [15]) Can a smooth projective variety be both defective and Grassmann defective?
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After describing a natural generalization of Terracini’s approach, we derive explicit numerical conditions in order to be Grassmann defective for a projective variety. A closer inspection to the case of rational normal scrolls makes it easy to produce infinite series of smooth varieties (even surfaces!) being both defective and Grassmann-defective, for instance: Example 1.6. For every h ≥ 4 the smooth rational scroll X(1, 2h + 2) ⊂ P2h+4 is both h-defective and (1, h)-defective. We work over an algebraically closed field K of characteristic zero. Indeed, our proof of Proposition 1.2 is characteristic free, but in the proof of Theorem 1.3 and in all related proofs we need char(K) = 0 because we quote [13]. We point out that in the case p := char(K) > 0 it would be sufficient to assume p > max{a, 4}, but we stress that we do not know any characteristic free proof of Theorem 1.3.
2
Preliminaries
Let Y be any algebraic variety, Z ⊆ Y a closed subscheme and D ⊂ Y an effective Cartier 2 divisor of Y . For any P ∈ Y let 2P or 2{P, Y } be the closed subscheme of Y with IP,Y as its ideal sheaf. Thus, if P ∈ Yreg , then length(2{P, Y }) = 1 + dim(Y ). The same notation will be used when P is an arbitrary integral closed subvariety of Y . The residual scheme ResD (Z) of Z with respect to D is the closed subscheme of Y with Hom(ID,Y , IZ,Y ) as the ideal sheaf. Thus ResD (Z) ⊆ Z. If P ∈ Dreg (and hence P ∈ Yreg ) we have 2{P, Y } ∩ D = 2{P, D} and ResD (2{P, Y }) = {P } (with its reduced structure). For any L ∈ Pic(Y ) we have an exact sequence on Y : 0 → IResD (Z),Y ⊗ L(−D) → IZ,Y ⊗ L → IZ∩D,D ⊗ (L|D ) → 0
(1)
From (1) we obtain the following very elementary form of Horace Lemma. Lemma 2.1. For any L ∈ Pic(Y ) we have h0 (Y, IZ,Y ⊗L) ≤ h0 (Y, IResD (Z),Y ⊗L(−D))+ h0 (D, IZ∩D,D ⊗ (L|D )) and h1 (Y, IZ,Y ⊗ L) ≤ h1 (Y, IResD (Z),Y ⊗ L(−D)) + h1 (D, IZ∩D,D ⊗ (L|D )). The next remark will be called the double residue trick. Remark 2.2. Fix a projective variety W , a closed subscheme T of W , a line bundle L ∈ Pic(W ), an effective Cartier divisor D on W , a point P ∈ Dreg with P ∈ / Tred . Let Z ⊂ W be a zero-dimensional scheme such that Zred = {P }. Set Z1 := Z. For any integer i ≥ 2 define inductively the scheme Zi by the formula Zi = ResD (Zi−1 ). Hence Zi ⊆ Zi−1 with strict inequality if Zi−1 6= ∅. Set ai := length(Zi ∩ D). The non-increasing
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sequence (a1 , . . . ) will be called the associated sequence of Z with respect to D. We have P length(Z) = i≥1 ai . Fix an integer i ≥ 1. By [4], Lemma 2.3 (see in particular Fig. 1 at p. 308) there is a scheme U ⊂ Z whose associated sequence is obtained from the associated sequence (a1 , . . . ) of Z omitting the term ai and a subscheme E ⊂ D with Ered = {P } and length(E) = ai such that to check h1 (W, IZ∪T ⊗L) = 0 (resp. h0 (W, IZ∪T ⊗L) = 0) it is sufficient to prove h1 (D, I(T ∩D)∪E,D ⊗ (L|D )) = h1 (W, IResD (T )∪U ⊗ L(−D)) = 0 (resp. h0 (D, I(T ∩D)∪E,D ⊗ (L|D )) = h0 (W, IResD (T )∪U ⊗ L(−D)) = 0). If a1 = 3, a2 = 2, a3 = 1, aj = 0 for j > 3 and i = 3, then we will call “ the (1, 3, 2)-trick ” this application of [4], Lemma 2.3.
Z=
U= (1,3,2)-trick E=
If a1 = 3, a2 = 2, a1 = 1, aj = 0 for j > 3 and i = 2, then we will call “ the (2, 3, 1)-trick ” this application of [4], Lemma 2.3.
Z=
U= (2,3,1)-trick E=
If a1 = 2, a1 = 1, aj = 0 for j > 2 and i = 2, then we will call “ the (1, 2)-trick ” this application of [4], Lemma 2.3.
Z=
U= (1,2)-trick -
E=
Now assume that we want to prove hi (W, IB∪T ⊗ L) = 0 (i = 0 or i = 1) for a “ general ” zero-dimensional scheme B with Bred = {P1 } and with associated sequence (3, 1, 0, . . . ), where P1 is a general point of a sufficiently general irreducible Cartier divisor D1 ; in our set-up, D1 will be a general fiber of the ruling. Set M := ResD1 (B). To prove the sought-for vanishing it is sufficient to prove hi (D1 , I(T ∩D1 )∪{P },D1 ⊗(L|D1 )) = hi (W, IT ∪M ⊗ L(−D1 )) = 0. This application of [4], Lemma 2.3, will be called “the (1, 3)-trick ”. And so on for other associated sequences. Example 2.3. Fix P ∈ D such that D is smooth at P , hence Y is smooth at P . Let Z ⊂ Y be a zero-dimensional scheme such that Zred = {P }. Assume Z curvilinear, i.e. assume that the Zariski tangent space of Z at P has dimension at most one. There is a germ C at P of a smooth curve containing Z. Let µ be the intersection multiplicity of C with D at P . Set a := length(Z). Hence Z is the divisor aP of C and hence OZ ∼ = OC /OC (−aP ). Set b := [a/µ] ≥ 0 and c = a − bµ. We have ai (Z) = µ for 1 ≤ i ≤ b, ab+1 (Z) = c and ai (Z) = 0 for all i ≥ b + 2. Remark 2.4. Let Y be an irreducible variety, L ∈ Pic(Y ) and V ⊆ H 0 (Y, L) a finitedimensional vector space. Assume V 6= {0}. For a general P ∈ Y we have V ∩H 0 (Y, I{P } ⊗ L) 6= V and hence dim(V ∩ H 0 (Y, I{P } ⊗ L)) = dim(V ) − 1. In several proofs, we are
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going to use this obvious remark without any further mention. By this observation, in all the statements of this paper we could assume “ mi ≥ 2 for every i ” without any loss of generality.
3
Ruled surfaces
Let Fe , e ≥ 0, be the Hirzebruch surface with invariant e, i.e. such that −e is the minimal self-intersection of a section of the ruling of Fe . Hence F0 ∼ = P1 × P1 is isomorphic to a smooth quadric surface and F1 is isomorphic to the blowing-up of P2 at one point. We fix a ruling f : Fe → P1 ; f is unique if and only if e > 0. We have Pic(Fe ) ∼ = Z⊕2 and we take, as a basis of Pic(Fe ), a section h of f with h2 = −e and a class, F , of the ruling f . Thus h · F = 1 and F 2 = 0. We will use both the additive and the multiplicative notation for line bundles and divisors on Fe . We have ωFe ∼ = −2h − (2 + e)F , and it is easy to 0 check h (Fe , OFe (ah + bF )) = 0 if a < 0. Moreover, by the projection formula we have La f∗ (OFe (ah + bF )) ∼ = i=0 OP1 (b − ie) for every a ≥ 0. Thus h0 (Fe , OFe (ah + bF )) = 0 if P a ≥ 0 and b < 0, h0 (Fe , OFe (ah + bF )) = t−1 − ie + 1) if 0 ≤ (t − 1)e ≤ b < te for some i=0 (b P 0 integer t with 0 < t ≤ a, h (Fe , OFe (ah+bF )) = ai=0 (b−ie+1) = (2b+2−ae)(a+1)/2 if a ≥ 0 and b ≥ ae−1 and h1 (Fe , OFe (ah+bF )) = 0 if a ≥ 0 and b ≥ ae−1. Now we will see how to translate any statement concerning the postulation of a zero-dimensional scheme Z ⊂ Fe in a statement concerning an interpolation problem for suitable polynomials (see Remark 3.1 for the case e = 0 and Remark 3.2 for the case e > 0). Remark 3.1. Fix integers α ≥ 0 and β ≥ 0. Since F0 ∼ = P1 × P1 , the vector space H 0 (F0 , OFe (αh + βF )) may be parametrized by the set of all bihomogeneous polynomials in the variables x0 , x1 , y0 , y1 whose monomials have degree α with respect to x0 , x1 and degree β with respect to y0 , y1 . Remark 3.2. Fix integers e > 0, a ≥ 0 and b. Take variables z0 , z1 and w and assign the La weight one to z0 and z1 and the weight e to w. Since f∗ (OFe (ah+bF )) ∼ = i=0 OP1 (b−ie), the vector space H 0 (Fe , OFe (ah + bF )) corresponds to the vector space of all weighted homogeneous polynomials in z0 , z1 and w with b as total weight degree and with degree ≤ a as a polynomial in w. Remark 3.3. Fix integers e ≥ 0, a ≥ 0, b ≥ ea and t ≥ 0. Fix A ⊂ R ⊂ Fe with R ∈ |F |, and A a zero-dimensional scheme such that length(A) = t. Since h1 (Fe , OFe (ah + (b − 1)F )) = 0, the restriction map ρ : H 0 (Fe , OFe (ah + bF )) → H 0 (R, OR (a)) is surjective. We have h0 (Fe , IA (ah+bF )) = h0 (Fe , OFe (ah+bF ))−min{t, a+1}, hence h1 (Fe , IA (ah+ bF )) = 0 if and only if t ≤ a + 1. Lemma 3.4. Let Z ⊂ F0 be the general union of 5 (resp. 6) double points. Then the restriction map H 0 (F0 , OF0 (3h + 3F )) → H 0 (Z, OZ (3h + 3F )) is surjective (resp. injective).
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Proof 3.5. We will do only the surjectivity part, the injectivity part being similar and easier. Let T ⊂ F0 be a general element of |h + 2F |. Hence T ∼ = P1 . Notice that T ·(3h+3F ) = 9 and h1 (F0 , OF0 (2h+F )) = 0. Hence the restriction map H 0 (F0 , OF0 (3h+ 3F )) → H 0 (T, OT (3h + 3F )) is surjective. Let Z ⊂ F0 and union of 5 double points with Zred ⊂ T . Thus length(Z ∩ T ) = 10. By Horace Lemma we have hi (F0 , IZ (3h + 3F )) = hi (F0 , IZred (2h + F )), i = 0, 1. Since there is a unique C ∈ |2h + F | containing Zred , we are done. ¤ Proof of Proposition 1.2. If 3x ≤ (a + 1)(b + 1) − 1 we need to prove the surjectivity of ρZ,a,b . If 3x > (a + 1)(b + 1) − 1 we need to prove the injectivity of ρZ,a,b . We will only do the case 3x ≤ (a + 1)(b + 1) − 1, the case 3x > (a + 1)(b + 1) − 1 requiring only notational modifications. Hence we assume 3x ≤ (a + 1)(b + 1) − 1. Just taking a suitable union of more double points, it is sufficient to prove the case (a+1)(b+1)−3 ≤ 3x ≤ (a+1)(b+1)− 1. We will assume that these inequalities are satisfied. If a = b = 3, then x = 5 and the result follows from Lemma 3.4. Hence we may assume a + b ≥ 7 and, say, b ≥ a. Suppose, for the moment, b ≥ 5. Thus, by induction on the integer a + b, we may assume that the result is true for the pair (a, b−1) and the pair (a, b−2), b ≥ 5. Fix D ∈ |F |. First assume a odd. Let W ⊂ F0 be a general union of x−(a+1)/2 double points of F0 . Let Z 0 ⊂ F0 be a general union of (a+1)/2 double points of F0 with (Z 0 )red ⊂ D. Thus ResD (Z 0 ) = (Z 0 )red and length(Z 0 ∩ D) = a + 1 = h0 (D, OD (ah + bF )). By Horace Lemma 2.1 it is sufficient to show h1 (F0 , IW ∪(Z 0 )red (ah + (b − 1)F )) = 0. By the inductive assumption and the inequality 3(x − (a + 1)/2) ≤ (a + 1)b − 1 we have h1 (F0 , IW (ah + (b − 1)F )) = 0. Notice that (a + 1)/2 + 3(x − (a + 1)/2) = 3x − (a + 1)(b + 1) + (a + 1)b. Hence to show that h1 (F0 , IW ∪(Z 0 )red (ah + (b − 1)F )) = 0 and the generality of (Z 0 )red it is sufficient to show that h0 (F0 , IW ∪D (ah + (b − 1)F )) ≤ h0 (F0 , IW (ah + (b − 1)F )) − (a + 1)/2. Since h0 (F0 , IW ∪D (ah+(b−1)F )) = h0 (F0 , IW (ah+(b−2)F )), it is sufficient to use the inductive assumption. Indeed, in all cases with 3x ≥ (a + 1)(b + 1) − 3, the inductive assumption implies h0 (F0 , IW (ah+(b−2)F )) = 0. Now assume a even. Let A ⊂ F0 be a general union of x − a/2 − 1 double points of F0 . Take a general union B of a/2 double points of F0 with support on D and a general P ∈ D. Apply the double residue trick (Remark 2.2) with respect to P . In order to conclude, it is sufficient to prove h1 (F0 , IA∪Bred ∪2{P,D} (ah + (b − 1)F )) = 0. Indeed, we have h1 (F0 , I2{P,D} (ah+(b−1)F )) = 0 because a > 0 and b−1 > 0, and h1 (F0 , IBred ∪2{P,D} (ah + (b − 1)F )) = 0 because h1 (F0 , I2{P,D} (ah + (b − 1)F )) = 0, h0 (F0 , I2{P,D} (ah + (b − 1)F )) ≥ card(Bred ) and Bred is general in D. Hence we see that h1 (F0 , IA∪Bred ∪2{P,D} (ah + (b − 1)F )) = 0 because h1 (F0 , IBred ∪2{P,D} (ah + (b − 1)F )) = 0, h0 (F0 , IBred ∪2{P,D} (ah + (b − 1)F )) = (a + 1)b − card(Bred ) − 3 ≥ card(A) and A is general in F0 . Consider now the case b = 4. We can restrict our attention to 3 ≤ a ≤ b (in fact, if a > b = 4 then we can apply the previous argument replacing a by b and b by a). Thus the surjectivity of the map ρZ,a,b is given by h1 (F0 , IZ (ah + 4F )) = 0 where Z = ∪xi=1 2Pi with 5a + 2 ≤ 3x ≤ 5a + 4 and a ≤ 4. By the same argument of Lemma 3.4 or by a direct computation (for example with a computer algebra system) we see that the systems
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IP6i=1 2Pi (3h + 4F ) and IP8i=1 2Pi (4h + 4F ) are non-special and the claim follows.
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¤
3 3 +s2 Proof of Theorem 1.3. Let Z = ∪si=1 3Pi ∪si=s 2Pi be a zero-dimensional scheme of 3 +1 fat points, where s3 and s2 are respectively the number of triple points and double points. By Laface we can easily treat the cases s2 = 0 or s3 = 0. In fact, if s2 = 0, then Z is the union of triple points and IZ (ah + bF ) is never special if a, b ≥ 4. When s3 = 0, Z is the union of double points. In this situation IZ (ah + bF ) is never special if a, b ≥ 3. Notice that we will never be able use induction on the parameters appearing in the statement of Theorem 1.3. We will just try to control the postulation of a certain zerodimensional scheme Γ in finitely many steps; if at one step, we cannot go on, then we failed. If all the steps can be done and Γ is exhausted (resp. the line bundle OF0 (ah + bF ) is exhausted), then we obtain h1 (F0 , IΓ (ah + bF )) = 0 (resp. h0 (F0 , IΓ (ah + bF )) = 0). By semicontinuity we will obtain Theorem 1.3 for certain data s, mi , 1 ≤ i ≤ s, related to Γ. Even more, instead of Γ is some of the steps we will use a virtual scheme and apply Remark 2.2.
Step 1: First of all we study the case b = 4 and a arbitrary. We consider the case with s3 > 0 and s2 > 0 and we may assume 5(a + 1) − 2 ≤ 6s3 + 3s2 ≤ 5(a + 1) + 2 (all other cases are reduced to this one). We assume s3 ≥ 2. We fix a fiber A and two different points P, Q ∈ A. We use 3P and the (2, 3, 1)-trick with respect to Q. Since length(Z ∩ A) = 3 + 2 we have hi (IZ∩A (4F )) = 0, for i = 0, 1. Thus hi (IZ (ah + 4F )) = hi (IResA (Z) ((a − 1)h + 4F )). The residual scheme Z 0 := ResA (Z) has s3 − 1 triple points, s2 double points, another double points 2P and a virtual scheme Q with layers (3, 1). Moreover, we observe that P and Q are general in A. Since on A we have again a virtual scheme of length 2 + 3 = 5, then hi (IZ 0 ∩A (4F )) = 0, for i = 0, 1 and hi (IZ (ah + 4F )) = hi (IZ 00 ((a − 2)h + 4F )). Z 00 := ResA Z 0 consists of s3 − 2 triple points, s2 double points and 2 simple points. If s3 is an even number we apply the previous argument s3 times and we obtain i h (IZ (ah + 4F )) = hi (IZ (s3 ) ((a − s3 )h + 4F )), where Z (s3 ) = N ∪ M with N union of s2 double points and M union of s3 simple points. One big inductive step consists in the reduction of an assertion on |(a−x)h+4F |, where x is an even integer with 0 ≤ x ≤ s3 −2, to an assertion on |(a − x − 2)h + 4F | by making two smaller inductive steps, first from |(a−x)h+4F | to |(a−x−1)h+4F | and then from |(a−x−1)h+4F | to |(a−x−2)h+4F |, each time using A ∈ |h|. In this part of the proof, for every big inductive step we take a different Ai general in |h|, 1 ≤ i ≤ s3 /2. Since the points in N are general, from [26] it follows that h1 (F0 , IN ((a − s3 )h + 4F )) = 0; we also have h0 (F0 , IN ((a − s3 )h + 4F )) ≥ s2 . This inequality would be sufficient to obtain h1 (F0 , IN ∩M ((a−s3 )h+4F )) = 0 if the points in M were general. Unluckily this is not the case, but the only restriction on M is the existence of s3 /2 general |h|, 1 ≤ i ≤ s3 /2, such that card(Ai ∩ M ) = 2 for all i; in particular, the points in Mi := M ∩ Ai are general in Ai . Define M≤i := ∪ij=1 Mj and set M≤0 := ∅. We claim that if h1 (F0 , IN ∩M≤i ((a−s3 )h+4F )) = 0 for some i ∈ {0, . . . , s3 /2}, then also h1 (F0 , IN ∩M≤i+1 ((a − s3 )h + 4F )) = 0. Indeed, the claim follows from the inequality h0 (F0 , IN ∩M≤i ((a − s3 )h + 4F )(−Ai+1 )) ≥ 2 and the fact that Mi+1 is general
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in Ai+1 . From this claim we inductively obtain h1 (F0 , IZ (s3 ) ((a − s3 )h + 4F )) = 0, as desired. If s3 is an odd number we apply the previous argument s3 − 1 times. Then we apply an Horace method using the last triple point and a double point. Again, the length on 0 A is 3 + 2 = 5 and we obtain hi (IZ (ah + 4F )) = hi (IZ (s3 )0 ((a − s3 )h + 4F )), where Z (s3 ) consists of s2 double points and s3 simple points. Finally, when s3 = 1 we just apply an Horace method using the triple point and a 0 double point obtaining again hi (IZ (ah + 4F )) = hi (IZ (1)0 ((a − 1)h + 4F )), where Z (1) consists of s2 double points and 1 simple points. If a ≥ s3 then Z has maximal rank for bidegree (a, 4). Moreover, by the assumption 6s3 + 3s2 ≤ 5(a + 1) + 2, we see that a < s3 only if a ≤ 10. Step 2: Fix an effective divisor R ∈ |F | and P1 ∈ R. We assume the existence of an integer k such that 1 ≤ k ≤ s and integers ji , 1 ≤ i ≤ k, 1 ≤ ji ≤ s, j1 = 1, Pk ji 6= jr if i 6= r, such that i=1 mji = a + 1. Choose Pji ∈ R, with Pji 6= Pjr if i 6= r. Choose general Pj ∈ F0 for j 6= {j1 , . . . , jk }. Set Z1 := ∪ki=1 mji Pji and Z2 = ∪j6={j1 ,...,jk } mj Pj . By semicontinuity it is sufficient to show h1 (F0 , IZ1 ∪Z2 (ah + bF )) = 0. Set Z 0 := ∪ki=1 (mji − 1)Pji ∪ Z2 . Thus Z 0 = ResR (Z1 ∪ Z2 ). By Horace Lemma we have hi (F0 , IZ1 ∪Z2 (ah + bF )) = hi (F0 , IZ 0 (ah + (b − 1)F )) i = 0, 1 and hence we may continue taking again R as Cartier divisor and trying to insert some of the points Pj , j 6= {j1 , . . . , jk }, into R. Notice that the set (Z2 )red is general in F0 , but that the points Pj1 , . . . , Pjk are not general in F0 . Step 3: Now assume that there are no such integers 1 ≤ k ≤ s, ji , 1 ≤ i ≤ k, 1 ≤ ji ≤ s, P j1 = 1, ji 6= jr if i 6= j, such that ki=1 mji = a + 1. We take the integers k and ji so that Pk Pk i=1 mji ≤ a + 1 and i=1 mji is maximal under this restriction. We claim that we may apply the double residue trick (Remark 2.2) with respect to many Pj with j 6= {j1 , . . . , jk } so that the total sum of the new virtual intersection with R is again a + 1. If k = s, then we cannot do that, but in this subcase we immediately get h1 (F0 , IZ (ah + bF )) = 0 by Horace Lemma and the assumption mi ≤ 3 ≤ b + 1. P If k < s by the maximality of ki=1 mji and the assumption mi ≤ 3 for all i we have a−1≤
k X
mji ≤ a.
i=1
If outside R we still have a fat point with multiplicity three, then we may always P P increase by one (case ki=1 mji = a) or by two (case ki=1 mji = a − 1) as we want the length of the restriction to R of a subscheme by using Remark 2.2 and respectively the (1, 3, 2)-trick and the (2, 3, 1)-trick. If outside R we have a double point, then we use the (1, 2)-trick to increase by one the length on R and we specialize it to a double point 2Q, Q general in R, to increase by two the length on R. In each step we apply the double residue trick only at one point because mi ≤ 3. Furthermore, at each step we need to consider at most one scheme
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having support on R and coming from a double residue trick of some previous steps, because mi ≤ 3 for every i (and, indeed, if there is any such scheme, then it comes from the previous step). More in detail we obtain the following situations. Let k3 and k2 be respectively the number of triple points and double points such that k = k2 + k3 . P • Case ki=1 mji = a − 1 i) If (k3 = s3 ) or (k3 ≤ s3 and k2 < s2 ) then we specialize a double point Q to R. The residual scheme consists of s3 − k3 triple points, s2 + k3 − k2 − 1 double points, k2 + 1 simple points. ii) If (k3 < s3 and s2 = k2 ) then we use the (2, 3, 1)-trick on a triple point Q. The residual scheme consists of s3 − k3 − 1 triple points, s2 + k3 − k2 double points, k2 simple points and a virtual scheme with layers (3, 1). P • Case ki=1 mji = a iii) If (k3 = s3 ) or (k3 ≤ s3 and k2 < s2 ) then we use the (1, 2)-trick on a double point Q. The residual scheme consists of s3 − k3 triple points, s2 + k3 − k2 − 1 double points, k2 simple points and a virtual scheme with layers (2). iv) If (k3 < s3 and s2 = k2 ) then we use the (1, 3, 2)-trick on a triple point Q. The residual scheme consists of s3 − k3 − 1 triple points, s2 + k3 − k2 double points, k2 simple points and a virtual scheme with layers (3, 2). Step 4: We use the set-up of Remark 2.2. If we apply the (1, 3)-trick or the (1, 3, 2)trick (resp., (1, 2)-trick, (2, 3)-trick, or (1, 2)-trick) at the next step (i.e. for the integer b0 := b − 1) we obtain a connected virtual scheme whose intersection with R has length larger by two (resp., by one) than the one of the original scheme. If P ∈ R and m > 0, ResR (mP ) = (m − 1)P (with the convention 0P = ∅), hence length(R ∩ ResR (mP )) = m − 1 = length(R ∩ mP ) − 1. Since in our set-up we may always take m ≤ 3 and a + 1 ≥ 4 · 3, we are always sure that at the next step we have a virtual scheme whose intersection with R has length at most a + 1. It follows that we can always apply the general machinery of Step 3 also with the integer b0 < b. Step 5: After b − 4 steps we arrive at b0 = 4. We are going to mimic the proof of the case b = 4 given in Step 1 with suitable modifications. In the present case we have a scheme Z = Z1 ∪ Z2 ∪ Z3 with Z1 union of general fat points with multiplicity 1, 2 or 3; Z2 union of general fat points with multiplicity 1 or 2, supported by points of R; Z3 either the empty set or the disjoint union of at most 2 schemes each of which belong to the following list: (2, 0, . . .); (3, 1, 0, . . .); (3, 0, . . .); (3, 2, 0, . . .), where we have specified the length of the intersection with R obtained via residual schemes. Furthermore, we have length(Z2 ∪ Z3 ) ≤ 2a − 1 and length(R ∩ (Z2 ∪ Z3 )) ≤ a; notice also that length(A ∩ (Z2 ∪ Z3 )) ≤ 2 for any A ∈ |h| and length(A ∩ (Z2 ∪ Z3 )) = 2 if and only if at the point A ∩ R the scheme Z2 ∪ Z3 is either a double point or a point of type (3, 1, 0, . . .) with respect to R. If length(A ∩ (Z2 ∪ Z3 )) = 2, then we insert a triple point on A. Let U be the union of all Ai ∈ |h| such that length(Ai ∩ (Z2 ∪ Z3 )) = 2. Since outside U ∩ (Z2 ∪ Z3 )) the scheme Z2 ∪ Z3 (not just its reduction) is contained in R, it follows that E := ResU (Z2 ∪ Z3 ) ⊆ R.
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We claim that if h1 (F0 , IS (eh + 4F )) = 0 for a suitable scheme S ⊆ Z1 ∪ Z2 ∪ Z3 and a suitable e ≤ a, then h1 (F0 , IS∪E (eh + 4F )) = 0. Indeed, by [13] applied to the linear system induced by H 0 (F0 , IS (eh + 4F )) on R, for general Ered we have h0 (F0 , IS∪E (eh + 4F )) = max{0, h0 (F0 , IS (eh + 4F )) − h0 (F0 , IS (eh + 4F )(−R))}. Hence in order to check h1 (F0 , IS∪E (eh + 4F )) = 0 it is sufficient to prove that h0 (F0 , IS (eh + 4F )(−R)) ≤ h0 (F0 , IS (eh + 4F )) − length(E). Since OF0 (eh + 4F )(−R) = OF0 ((e − 1)h + 4F ), we get the above inequality by a standard inductive argument (we refer to the proof of Proposition 1.2 and to Step 1 for more details in a similar situation). If length(A ∩ (Z2 ∪ Z3 )) = 1, then the connected component C of Z2 ∪ Z3 with Cred = A ∩ R is contained in R. Set λ := length(C) and notice that λ ≤ 3. If λ = 1, then we insert a triple point on A and simultaneously apply the (1, 3, 2)-trick. After this small step, there is a length 5 scheme on A (a length 2 connected component comes from the triple point, a length 3 connected component comes from the (1, 3, 2)-trick). By applying Horace Lemma 2.1 we reduce to a case with a length 3 scheme on A (one reduced point coming from the triple point and one length 2 connected component coming from the (1, 3, 2)-trick). Then either we apply a (2, 3, 1)-trick or (if no triple point is left) we insert on A a double point. Assume now λ = 2. In this case, we insert a triple point on A and simultaneously apply the (1, 2, 3)-trick. After this small step, there is a length 5 scheme on A, hence by applying Horace Lemma 2.1 we can reduce to a case with on A a connected scheme of length 1 (coming from the triple point) and a connected scheme of length 3 (coming from the (1, 2, 3)-trick). Then we apply either the (1, 3, 2)-trick or (if no triple point is left) the (1, 2)-trick. Finally, in the case λ = 3, we insert a triple point on A and simultaneously apply the (1, 2, 3)-trick. After this small step, there is a length 5 scheme on A (one reduced component coming from C, one length 2 connected component coming from the triple point and another one of length 2 coming from the (1, 2, 3)-trick). By Horace Lemma 2.1 we reduce to a case with a length 5 scheme on A (one reduced point coming from C, another reduced point coming from the triple point and one length 3 connected component coming from the (1, 2, 3)-trick). After applying Horace Lemma 2.1 once again, the proof is over. ¤ Next we classify all exceptional cases on F0 with mi ≤ 3 for all indices i, mi = 2 for at least one index i and mi = 3 for at least one index i (see [26] for the other exceptional cases with mi ≤ 3). Proposition 3.6. Fix integers b ≥ 2, u > 0, c > 0. take u + c general points Pi ∈ F0 , 0 1 ≤ i ≤ u + c, and set Z := ∪ui=1 3Pi ∪ ∪u+c j=u+1 2Pi . We have h (F0 , IZ (2h + bF )) · h1 (F0 , IZ (2h + bF )) 6= 0 if and only if b = 2u + c − 1 and u + c − 1 ≡ 0 (mod 2). Proof 3.7. Set W := ∪u+c i=1 2Pi . Apply Horace Lemma u times with respect to the divisors Ri ∈ |OF0 (0, 1)| such that Pi ∈ Ri , 1 ≤ i ≤ u. Since W = ResR1 ∪···∪Ru (Z), we obtain both h0 (F0 , IZ (2h+bF )) = h0 (F0 , IW (2h+(b−u)F )) and h1 (F0 , IZ (2h+bF )) = h1 (F0 , IW (2h+ (b − u)F )). Thus we restrict to analyze the homogeneous system IW (2h + (b − u)F ). By [26], this system is special if it has the form IT (2h + (2d)F ) with T = ∪2d+1 i=1 2Pi . This can
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happen only if we require 2d = b − u and 2d + 1 = u + c or equivalently, b = 2u + c − 1 and u + c − 1 ≡ 0 (mod 2). The proof of Theorem 1.3 may be applied to many other cases. We give, for example, the following result. P P Theorem 3.8. Fix integers x ≥ 0, cj > 0, 1 ≤ j ≤ x, b ≥ 4 + xj=1 cj a ≥ 11 + xj=1 cj , s > 0 and mi , 1 ≤ i ≤ s, such that 1 ≤ mi ≤ 3 for every i. Fix s + x general points Pi , Qj ∈ F0 , 1 ≤ i ≤ s, 1 ≤ j ≤ x and set Z := ∪si=1 mi Pi ∪ ∪xj=1 cj Qj . Then the restriction map ρZ,a,b : H 0 (F0 , OF0 (ah + bF )) → H 0 (Z, OZ (ah + bF )) has maximal rank. Proof 3.9. The case x = 0 is just Theorem 1.3. Fix R ∈ |F | and take as Qj x general P points of R. Hence ResR (∪xj=1 cj Qj ) = xj=1 (cj − 1)Qj (with the convention 0Pj = ∅) and length(R ∩ (∪xj=1 cj Qj )) = c1 + · · · + cx . Then we repeat the proof of Theorem 1.3 taking P this fiber R to apply Horace Lemma (see the details of Step 4). After at most xj=1 cj steps we have exhausted all fat points with multiplicity ≥ 4, hence we are again in the set-up of Theorem 1.3. We never use the double residue trick with respect to any point Qj , but only with respect to the other s points. Hence in our new situation the same bounds apply. Our method can also be applied in several situations in which the points are not general. In this way one obtains results for complete linear systems on a (non-general) blown - up surface. Here we give only a very easy example. The interested reader may construct in the same way thousands of other examples. Following the proof of Theorem 1.3 it is easy to construct examples with maximal rank. Example 3.10. Fix integers e ≥ 2, b > e and y such that 1 ≤ y ≤ b. Fix y distinct points Pi ∈ Fe \h such that f (Pi ) = 6 f (Pj ) if i 6= j and set Z := ∪yi=1 2Pi , T := ∪yi=1 f −1 (f (Pi )). We have h0 (Fe , OFe (h+zF )) = z+1 if 0 ≤ z ≤ e−1, while h0 (Fe , OFe (h+zF )) = 2z−e+2 if z ≥ e. Now assume y > b − e and 2y ≤ b. Since ResT (Z) = ∪yi=1 {Pi }, every m ∈ H 0 (Fe , IZ (h+bf )) vanishes on T , i.e. h0 (Fe , IZ (h+bF )) = h0 (Fe , I∪si=1 {Pi } (h+(b−y)F )) > h0 (Fe , OFe (h + bF )) − 3y. The proofs of Theorem 1.3 and Theorem 3.8 give the following result. Theorem 3.11. Fix integers e > 0, s > 0, 1 ≤ i ≤ s, mi , a ≥ 11 and b such that P b ≥ 4 + ae > 0, 1 ≤ mi ≤ 3 for every i and si=1 mi (mi + 1)/2 ≤ (b − ae − 1)(a + 1). Fix general Pi ∈ Fe and set Z := ∪si=1 mi Pi . Then h1 (Fe , IZ (ah + bF )) = 0. Indeed, just use |F | as a ruling and take the curves A, Ai in |h + eF | instead of |h|; notice however that the prescribed bound is certainly not sharp for e > 0. We also obtain the following result.
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Proposition 3.12. Fix integers e > 0, s > 0, a > 0, b ≥ ae, mi , 1 ≤ i ≤ s. Set P δ := max1≤i≤s {mi } and assume si=1 mi (mi + 1)/2 ≤ (b − ea − 1)(a + 1 − δ). Fix general Pi ∈ Fe , 1 ≤ i ≤ s, and set Z := ∪si=1 mi Pi . Then h1 (Fe , IZ (ah + bF )) = 0. Proof 3.13. Fix an effective divisor R ∈ |F |. We take the integers k and ji , 1 ≤ i ≤ k, Pk Pk mji ≤ a + 1 and so that i=1P i=1 mji is maximal with this restriction. We have k a − δ + 1 ≤ i=1 mji ≤ a + 1 by the definition of δ. Choose Qji ∈ R, with Qji 6= Qjr if i 6= r. Choose general Oj ∈ Fe for j 6= {j1 , . . . , jk }. Set Z1 := ∪ki=1 mji Qji and Z2 = ∪j6={j1 ,...,jk } mj Oj . By semicontinuity it is sufficient to show h1 (F0 , IZ1 ∪Z2 (ah + bF )) = 0. Set Z 0 := ∪ki=1 (mji − 1)Qji ∪ Z2 . Thus Z 0 = ResR (Z1 ∪ Z2 ). By Horace Lemma and the P inequality ki=1 mji ≤ a + 1 it is sufficient to prove h1 (Fe , IZ 0 (ah + (b − 1)F )) = 0. In order to do so, we may continue taking again R as Cartier divisor and specializing onto P R the support of some of the fat points mj Oj , j 6= {j1 , . . . , jk }. Since si=1 mi (mi + 1)/2 ≤ (b − ea − 1)(a + 1 − δ), after at most b − ea − 1 steps we have inserted all zerodimensional schemes. Hence with another step the claim is reduced to the vanishing of h1 (Fe , OFe (ah + zF )) for some z ≥ b − ea − 1 and the proof is over. The same proof gives the following result. Proposition 3.14. Fix integers s > 0, b ≥ a > 0, mi > 0, 1 ≤ i ≤ s. If s = 1, then set δ := 0 and assume m1 ≤ a + 1. If s ≥ 2, then set δ := max2≤i≤s {mi } and assume P δ ≤ m1 ≤ a + 1. Assume si=1 mi (mi + 1)/2 ≤ (b + 1)(a + 1 − δ). Fix general Pi ∈ F0 , 1 ≤ i ≤ s, and set Z := ∪si=1 mi Pi . Then h1 (F0 , IZ (ah + bF )) = 0. Finally we turn to more general ruled varieties. Let C be a smooth projective curve of genus g ≥ 0 and E a rank n vector bundle on C. Set X := P(E) and call f : X → C the natural projection. If L ∈ Pic(X), then there is a unique line bundle R on C such that L ∼ = OX (t) ⊗ f ∗ (R). Assume h1 (C, E ⊗ R) = 0 and set τ (L) the maximal integer b ≥ 0 such that h1 (C, E ⊗ R(−bP )) = 0 (i.e. h0 (C, E ⊗ R(−bP )) = h0 (C, E ⊗ R) − bn) for a general P ∈ C. The proof of Proposition 3.12 gives verbatim the following results. Theorem 3.15. Fix a ruled surface X = P(E) and L ∈ Pic(X) with L ∼ = OX (a)⊗f ∗ (R). Assume a ≥ 11 and h1 (C, E ⊗ R) = 0. Fix integers s > 0 and mi , 1 ≤ i ≤ s, such that 1 ≤ mi ≤ 3. Choose general Pi ∈ X, 1 ≤ i ≤ s, and set Z = ∪si=1 mi Pi . Assume Ps 1 i=1 mi (mi + 1)/2 ≤ (a + 1)τ (L). Then h (X, IZ ⊗ L) = 0. Proposition 3.16. Fix a ruled surface X = P(E) and L ∈ Pic(X) with L ∼ = OX (a) ⊗ ∗ 1 f (R). Assume a > 0 and h (C, E ⊗ R) = 0. Fix integers s > 0 and mi > 0, 1 ≤ i ≤ s, and general Pi ∈ X, 1 ≤ i ≤ s. Set Z := ∪si=1 mi Pi . If s = 1, set δ = 0 and assume m1 ≤ a + 1. If s ≥ 2, set δ := max2≤i≤s {mi }. Assume δ ≤ m1 ≤ a + 1 − δ and Ps 1 i=1 mi (mi + 1)/2 ≤ (a + 1 − δ)τ (L). Then h (X, IZ ⊗ L) = 0.
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Remark 3.17. Let X ⊂ Pr be an integral variety and P1 , . . . , Ph general points of X. By the characteristic free part of Terracini’s lemma ([1], part (1) of Corollary 1.11, or [18]) if ρZ,1,r has maximal rank, then X is not h-defective, the converse being in general true only in characteristic zero. Hence Horace method can be used to prove the non h-defectivity of certain projective varieties even in positive characteristic.
4
Open questions
In this section we raise two questions concerning the postulation of general unions of fat points on a scroll over a curve. Let C be a smooth projective curve of genus g ≥ 0 and E a rank n vector bundle on C. Set X := P(E) and call f : X → C the natural projection. Let OX (1) the tautological quotient line bundle on X, i.e. the only line bundle on X such that f∗ (OX (1)) ∼ = E. n−1 The restriction of OX (1) to any fiber F ∼ of f has degree one. We have Pic(X) ∼ =P = L ∗ ∗ ∼ f (Pic(C)) ZOX (1). For any R ∈ Pic(C) and any t ≥ 0 we have f∗ (OX (t) ⊗ f (R)) = S t (E) ⊗ R. For all integers i, x with 1 ≤ i ≤ n − 2 we have Ri f∗ (OX (x) ⊗ f ∗ (R)) = 0. For every integer b ≥ −n + 1 we have Rn−1 f∗ (OX (b) ⊗ f ∗ (R)) = 0. Remark 4.1. Fix positive integers n, m and t with n ≥ 2. If t ≥ m − 1, set µ(n, t, m) := ¡n+m−1¢ . Hence if t ≥ m − 1 then the integer µ(n, t, m) is the length of a fat point mP n of multiplicity m on any n-fold. Let X = P(E) be an n-dimensional scroll over a smooth curve and L ∈ Pic(X) such that L|F ∼ = OF (t) for any fiber F of the ruling of X. Fix any P ∈ X and let F be the fiber of the ruling of X. Since L|F ∼ = OF (t), if m ≥ t + 1, ¡n+t−1¢ then mP|F imposes at most n−1 independent conditions to H 0 (F, OF (t)). We have ¡ ¢ ¡n+m¢ ResF (mP ) = (m − 1)P . For any m ≥ t + 2 set µ(n, t, m) = (m − t − 1) n+t−1 + n . n−1 0 0 By the exact sequence (1) we have h (X, L) − h (X, ImP ⊗ L) ≤ µ(n, t, m) for any ndimensional scroll X = P(E) and any L ∈ Pic(X) such that L|F ∼ = OF (t) for any fiber F s of the ruling of X. Hence for any disjoint union Z = ∪i=1 mi Pi of s fat points of X we have P P h0 (X, L) − h0 (X, IZ ⊗ L) ≤ si=1 µ(n, t, mi ). The integer si=1 µ(n, t, mi ) will be called virtual number of conditions imposed by Z and we will say that Z imposes the expected P number of conditions to L if h0 (X, IZ ⊗ L) = max{0, h0 (X, L) − si=1 µ(n, t, mi )}. Question 4.2. Let f : X = P(E) → C be an n-dimensional scroll over a smooth curve C and L ∈ Pic(X) such that L|F ∼ = OF (t) for any fiber F of the ruling of X. Write OX (1) ∈ Pic(X) for the line bundle on X such that f∗ (OX (1)) ∼ = E and call R the unique ∗ ∼ line bundle on C such that L = OX (t) ⊗ f (R). Let β(L) be the maximal integer x ≥ 0 such that for x general points P1 , . . . , Px of C we have h0 (C, E ⊗ R(−P1 − · · · − Px )) = h0 (C, E ⊗ R) − xn. Next, recall from the previous Section that τ (L) denotes the maximal integer b ≥ 0 such that h0 (C, E ⊗ R(−bP )) = h0 (C, E ⊗ R) − bn for a general P ∈ C. Hence τ (L) ≤ β(L). (i) Is it true that a general union Z = ∪si=1 mi Pi of s fat points of X imposes the ¡ ¢ P or at least if expected number of conditions to L if si=1 µ(n, t, mi ) ≤ β(L) n+t−1 n−1
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¡ ¢ µ(n, t, mi ) ≤ τ (L) n+t−1 ? n−1 (ii) Fix E and the integer t. Is there an integer b(C, E, t) such that for all integers d ≥ b(C, E, t) and every R ∈ Picd (C) a general union of fat points imposes the expected number of conditions to OX (t) ⊗ f ∗ (R) ? Ps
i=1
We expect an affirmative answer to both parts of Question 4.2. Remark 4.3. Take X, t and L as in part (i) of Question 4.2, P ∈ X and set Z := mP , ¢ ¡ then the linear system |IZ ⊗ L| contains m > 0. If m > t + 1, i.e. if µ(n, t, m) 6= n+m−1 n−1 the fiber of the ruling of X passing through P with multiplicity m − t − 1. Hence if W ⊂ X is a general union of fat points whose expected number of conditions for L is not deg(W ), then |IW ⊗ L| has always a divisor in its base locus. Compare the previous remark with part (i) of the following question; in the planar case, part (i) is a conjecture of B. Segre, while part (ii) is a conjecture due independently to B. Harbourne and A. Hirschowitz; in the plane the two conjectures are equivalent ([14]). Question 4.4. Take e = 0, 1 and integers a, b and a general union Z = ∪si=1 mi Pi ⊂ Fe of finitely many fat points such that Z does not imposes the expected number of conditions to |ah + bF |. (i) Is it true that the linear system |IZ (ah + bF )| has an unreduced base component (even throwing away from the base components the fibers of the ruling arising by Remark 4.3 if mi > a + 1 for some i or if e = 0 and mi > b + 1 for some i) ? (ii) Is it true that the associated linear system L on the blowing - up of Fe at the points P1 , . . . , Ps has a (−1)-curve as multiple base component ? In Question 4.4 we assume e = 0, 1 because these are the only values of e such that the anticanonical line bundle −KFe is ample (and even very ample) and this condition seems to be very important (see the theory of nodal curve ([5], [16], Lemma 3, [32]) which is related to the case mi = 2 for all i). Since F1 is the blowing - up of P2 at one point, Question 4.4 for e = 1 follows from the corresponding conjecture (called the HarbourneHirschowitz conjecture) for P2 ([22], [25], [13]). As a matter of fact, in the planar case if P a system L := |dH − hi=1 mi pi | splits as L = N Γ + M, and if the strict transform C of Γ at the blowing up of P2 at the points P1 , . . . , Ph is a (−1)−curve, then by Riemann-Roch we have N dim(L) = dim(M) ≥ v(M) = v(L) + 2 If v(M) ≥ 0 and N ≥ 2 we say L is a (−1)−special system; the Harbourne-Hirschowitz Conjecture says “L is special if and only if is (−1)−special”. In the case of Hirzebruch surfaces we can still have the series of inequalities as in the planar case, but this time some problems can arise. In fact, it can happen that the
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system L on Fe seems to have intersection multiplicity −1 with the (−1)−curve E, but this is due to the curve h of self-intersection −e which ”hides” the real value of L · E. P Example 4.5. Consider on F6 the system L := |4h + 24F − 11 i=1 3Pi | (v(L) = −1) P11 and the (−1)−curve E := |h + 8F − i=1 Pi |; then we compute L · E = −1. The P 0 residual system is L0 := |3h + 16F − 11 i=1 2Pi | and we have L · h = −2 then we have P11 L − E − h = |2h + 16F − i=1 2Pi | = 2E then the system L splits as L = h + 3E and is not empty. Laface, in [26], solved this problem giving a different definition of (−1)−special system. For that, we need the following procedure: given a linear system L := |ah + bF − Ph i=1 mi Pi | on Fe : 1) if it does exist a (−1)−curve E such that −t := L · E < 0 then substitute L with L − tE and goto step 1), else goto step 2). 2) if L · h < 0 then substitute L with L − h and goto step 1), else finish. After a finite number of steps, we have a new linear system M, i.e., the residual linear system. Definition 4.6. A linear system L := |ah + bF − v(M) > v(L).
Ph i=1
mi Pi | on Fe is (−1)−special if
Obviously we can state again a modified Harbourne-Hirschowitz conjecture, but, this time, for the specialty of a linear system L such that L = N Γ + M it is not enough to have v(M) ≥ 0 and N ≥ 2, as the following example shows. P Example 4.7. If we consider on F4 the system L := |4h + 16F − 7i=1 3Pi | and the P (−1)−curve E := h + 5F − 7i=1 Pi , then we have L · E = −1, hence L splits as E + L0 P where L0 := |3h + 11F − 7i=1 2Pi |. Now, it is easy to see that L0 := h + |2h + 11F − Ph P − hi=1 2Pi | · E = −1, so we can conclude L = h + 2E + M, i=1 2Pi | and |2h + 11F P where M := |h + 6F − 7i=1 Pi |. If we look carefully at their virtual dimensions, we see that v(L) = v(M) = 2 > 0; then L is non special as shown by Laface in [26]. Instead, if we consider the old definition of (−1)−special system, as given by Harbourne and Hirschowitz, L is a counterexample to a possible extension of their conjecture on Hirzebruch surfaces Fe , at least for e > 2.
5
Higher dimensional scrolls
Let C be a smooth projective curve of genus g ≥ 0 and E a rank n vector bundle on C. Set X := P(E) and call f : X → C the natural projection. Fix P ∈ X and Z ⊂ X such that Zred = {P }. Let D := f −1 (f (P )) be the fiber of f over f (P ) and (a1 , . . . ) the associated sequence of Z with respect to D. We will say that Z has vertical length a1 . For an arbitrary zero-dimensional subscheme of X we may define in this way the vertical length of each of its connected components. We recall that if Z is curvilinear,
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then all except at most one of the non-zero entries of the sequence (a1 , . . . ) are equal to a1 (Example 2.3). The set of all curvilinear subschemes supported by P and with length a is parametrized by an irreducible and smooth variety of dimension (n − 1)(a − 1). The set of all curvilinear subschemes of Z supported by P , with length a and with vertical length a1 is parametrized by an irreducible variety. Remark 5.1. Fix P ∈ X and Z ⊂ X with Zred = {P } and vertical length a1 . For any ¢ ¡ ≤ a1 − 1 the scheme Z does not impose L ∈ Pic(X) whose fiber degree, t, satisfies n+t−1 n−1 length(Z) independent conditions to the complete linear system |L|, i.e. h0 (X, L) − h0 (X, IZ ⊗ L) < length(Z). The next result shows that quite often Remark 5.1 gives a sharp bound for curvilinear zero-dimensional schemes with high vertical length. Proposition 5.2. Let f : X = P(E) → C be a geometrically ruled n-dimensional manifold on the smooth curve C and L ∈ Pic(X). Let t > 0 be the degree of the restriction of L to any fiber of f . Let τ (L) be the maximal integer x ≥ 0 such that ¡ ¢ h0 (X, L(−xf −1 (P ))) = h0 (X, L) − x n+t−1 . Assume either n = 2 or char(K) = 0. Fix a n−1 ¡ ¢ general P ∈ X and integers a ≥ a1 > 0 such that n+t−1 ≥ a1 and a ≤ a1 τ (L). Then a n−1 general zero-dimensional curvilinear subscheme Z ⊂ X with Zred = {P }, length(Z) = a and with vertical length a1 satisfies h0 (X, L ⊗ Z) = h0 (X, L) − a. Proof 5.3. Set D := f −1 (f (P )) ∼ = Pn−1 and T := f −1 ((τ (L)f (P )). Hence T is the infinitesimal neighborhood of order τ (L)−1 of D in X, By assumption the restriction map H 0 (X, L) → H 0 (T, L|T ) is surjective. Hence it is sufficient to show that the restriction map H 0 (T, L|T ) → H 0 (Z, L|Z ) is surjective. Apply Horace Lemma 2.1 τ (L) times. By Remark 5.1 and the inequality a ≤ a1 τ (L) we see that it is sufficient to show that for a general degree a1 curvilinear subscheme W supported by one point of Pn−1 we have ¡ ¢ h1 (Pn−1 , IW (t)) = 0. Since n+t−1 ≥ a1 , this is obvious if n = 2 and true by [13] if n−1 n ≥ 3. Remark 5.4. Here we consider h0 (X, L), i.e. we consider the case mi = 0 for all i, that is to say Z = ∅. Even this case may be very complicated. If E ∼ = OC⊕n , then the study of h0 (X, L) for all R ∈ Pic(X) such that h0 (C, R) ≥ 2 and h1 (C, R) ≥ 2 is equivalent to the Brill - Noether theory of C. Hence if g ≥ 3 this study depends on the isomorphic classes of C, not just on its genus. Now assume E arbitrary but take R ∼ = OC . Hence we need 0 t to compute h (C, S (E)). If E is a general stable bundle on C (say with degree d and ¡ ¢ rank n), then h0 (C, S t (E)) = max{0, ((td/n) + 1 − g) t+n−1 ([6], Th. 0.1). t Proposition 5.5. Fix X = P(E), L ∼ = OX (1) ⊗ f ∗ (R), integers s > 0, mi > 0, 1 ≤ i ≤ s, and general Pi ∈ X, 1 ≤ i ≤ s. Set Z := ∪si=1 mi Pi and Qi := f (Pi ) ∈ C. Then P h0 (X, IZ ⊗ L) = max{0, h0 (C, E ⊗ R(− si=1 (mi − 1)Qi )) − s}.
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Proof 5.6. Set T := f −1 ((mi − 1)Qi ) (seen as an effective Cartier divisor of X) and Z 0 := ∪si=1 Pi (with its reduced structure). The restriction of L to any fiber of the ruling has degree 1, hence T is in the base locus of H 0 (X, IZ ⊗ R); notice moreover that ResT (Z) = Z 0 . By Horace Lemma we obtain h0 (X, IZ ⊗ L) = h0 (X, IZ 0 ⊗ L(−T )). Since P h0 (X, L(−T )) = h0 (C, E ⊗ R(− si=1 (mi − 1)Qi )), and the points Pi are general, we conclude. Now we assume g = 0 and give the promised example of higher dimensional scroll with an embedding of degree one on each fiber and for which many general unions of double points have not maximal rank. As in the case of surfaces (Example 3.10) this example arises only for unbalanced scrolls. Fix an integer n ≥ 2. For all integers a1 ≥ · · · ≥ an set X(a1 , . . . , an ) = P(OP1 (a1 ) ⊕ · · · ⊕ OP1 (an )) (or just X if there is no danger of misunderstandings) and let f : X(a1 , . . . , an ) → P1 be the ruling. Let OX(a1 ,...,an ) (1) be the line bundle such that f∗ (OX(a1 ,...,an ) (1)) ∼ = OP1 (a1 )⊕· · ·⊕OP1 (an ). Without changing the isomorphic class of X, but just considering OX ⊗ f ∗ (P1 (−a1 )) instead of OX , we may assume a1 = 0. Example 5.7. Fix integers 0 ≤ a2 ≤ · · · ≤ an and set X := X(0, a2 , . . . , an ) and OX (t, b) := OX (t) ⊗ f ∗ (OP1 (b)). The line bundle OX (t, b) is spanned if and only if t ≥ 0 and b > tan . The line bundle OX (t, b) is ample if and only if it is very ample if and only if t > ban . Fix integers b and y such that b > an , 0 < y ≤ b and ny > b + an and general P1 , . . . , Py ∈ X. Set Z := ∪yi=1 2Pi . Set T := f −1 (f (P1 )) ∪ · · · ∪ f −1 (f (Py )). By the generality of the points P1 , . . . , Py we have f (Pi ) 6= f (Pj ) for i 6= j. Hence T is union of y disjoint fibers of f and ResT (Z) = {P1 , . . . , Py }. For any P ∈ X if m ∈ H 0 (X, I2{P } (1, b)), then m|f −1 (f (P )) ≡ 0. Hence every m ∈ H 0 (X, IZ (1, b)) vanishes on T . By Horace Lemma and the generality of the points P1 , . . . , Py we have h0 (X, IZ (1, b)) = h0 (X, I{P1 ...,Py } (1, b − y)) = max{0, h0 (X, OX (1, b − y)) − y}. Since y ≤ b and ny > b + an , we have h0 (X, IZ (1, b)) 6= 0 and h1 (X, IZ (1, b)) 6= 0.
6
Rational normal scrolls and Grassmann defectivity
The systematic study of defective varieties goes back to the old Italian school: we wish to mention at least the contributions of Francesco Palatini ([27], [28]), Gaetano Scorza ([29], [30], [31]) and Alessandro Terracini ([33], [35]). This great amount of work was recently rediscovered by various authors, among whom Luca Chiantini and Ciro Ciliberto; we refer to their papers [8] and [9] for rigorous proofs and powerful generalizations of the classical results in the field. Here instead we focus on varieties which are Grassmann defective according to Definition 1.4. We recall from [19] the following results, which are essentially contained also in [34]. Proposition 6.1. Let X ⊂ Pr be an integral nondegenerate projective variety of dimen-
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sion n. Let σ : Pk × X → P(k+1)(r+1)−1 be the Segre embedding of Pk × X. Then X is (k, h)-defective with defect δk,h (X) = δ if and only if σ(Pk × X) is h-defective with defect δh (σ(Pk × X)) = δ. Lemma 6.2. Let X ⊂ Pr be an integral nondegenerate projective variety of dimension n. Let σ : Pk ×X → P(k+1)(r+1)−1 be the Segre embedding of Pk ×X. Fix p(0) , . . . , p(h) general (j) (j) points on X and λ(0) , . . . , λ(h) general points in Pk , so that P (j) := (λ0 p(j) , . . . , λk p(j) ) is a general point on σ(Pk × X) ⊂ P(k+1)(r+1)−1 for j = 0 . . . h; finally, take a general point P ∈< P (0) , . . . , P (h) >. Then there is a natural identification between: • hyperplanes H ⊂ P(k+1)(r+1)−1 such that TP (Sech (σ(Pk × X))) ⊂ H; • k-dimensional linear systems H of hyperplane sections of X ⊂ Pr with a projectivity ω : H → Pk such that all the elements of the linear system pass through the points p(j) ∈ X and for every j the hyperplane section of the linear system corresponding to λ(j) is tangent to X at p(j) . From the previous facts it is rather easy to deduce sufficient conditions to be Grassmann defective for a projective variety. Theorem 6.3. Let 0 ≤ k ≤ 1 and h > k be integers. Let X ⊂ Pr be a nondegenerate integral projective variety. Let V ⊆ H 0 (X, OX (1)) such that Pr = PV . Assume that there is a Cartier divisor D on X such that dim PH 0 (X, OX (D)) ≥ h + 1
(2)
(k + 1) dim PV (−D) ≥ h + r(k + 1) − min{(n + k + 1)(h + 1) − 1, r(k + 1) + k} + δ.
(3)
Then X is (k, h)-defective with defect at least δ. Proof 6.4. Fix p(0) , . . . , p(h) general points in X and λ(0) , . . . , λ(h) general points in Pk (if k = 0 of course we have λ(0) = . . . = λ(h) ). By (2) there is an effective divisor linearly equivalent to D passing through p(0) , . . . , p(h) ; with a slight abuse of notation we denote such a divisor by D. Moreover, we claim that (3) yields c := r(k + 1) + k − min{(n + k + 1)(h + 1) − 1, r(k + 1) + k} + δ independent k-dimensional linear systems of hyperplane sections of X with base locus D and moving parts passing through p(0) , . . . , p(h) in correspondence with the coefficients λ(0) , . . . , λ(h) . Indeed, if k = 1, since λ(0) , . . . , λ(h) are general, such a condition says exactly that the generic fiber of the natural map G(P1 , PV (−D)) −→ (P1 )h+1 /Aut(P1 ) which associates to a one-dimensional linear system the (h + 1)-ple of coefficients corresponding to its divisors through p(0) , . . . , p(h) , must have dimension at least c − 1. Therefore the previous condition is satisfied if dim G(P1 , PV (−D)) − (h + 1) + 3 ≥ c − 1,
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i.e. 2 dim PV (−D) ≥ h + c − 1. If instead k = 0, the claimed condition may be rephrased as follows: dim PV (−D)(−p(0) . . . − p(h) ) ≥ c − 1 and, since p(0) , . . . , p(h) are general, this means dim PV (−D) ≥ h + c. Summing up, for k = 0, 1 we obtain (k + 1) dim PV (−D) ≥ h + c − k, which is nothing but (3). So the claim is checked and by applying Lemma 6.2 we deduce that TP (Sech (σ(Pk × X))) is contained in at least c independent hyperplanes of Pr(k+1)+k ; hence dim Sech (σ(Pk × X)) ≤ r(k + 1) + k − c and σ(Pk × X) is h-defective with defect at least δ. Now the thesis directly follows from Proposition 6.1. The previous result becomes quite effective whenever one has a good control on the geometry of the linear series on X. Corollary 6.5. Fix integers 0 ≤ k ≤ 1, h > k, and δ ≥ 1. Let 0 < a1 ≤ . . . ≤ an be positive integers and let X = X(a1 , . . . , an ) ⊂ Pa0 +...+an +n−1 be the rational normal scroll of type (a1 , . . . , an ). Assume that there exists a nonnegative integer b such that either b+1 ≥ h+2 (k + 1)
n X
Ã
max{ai − b + 1, 0} ≥ k + 1 + h + (k + 1)
i=1
n X
! ai + n − 1
i=1
− min{(n + k + 1)(h + 1) − 1, Ã n ! X (k + 1) ai + n − 1 + k} + δ i=1
or n X
max{ai − b + 1, 0} ≥ h + 2
i=1
(k + 1)(b + 1) ≥ k + 1 + h + (k + 1)
à n X
! ai + n − 1
i=1
− min{(n + k + 1)(h + 1) − 1, Ã n ! X (k + 1) ai + n − 1 + k} + δ. i=1
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Then X is (k, h)-defective with defect at least δ. Proof 6.6. Let H be a divisor associated to the tautological sheaf OX (1) and F be a divisor associated to the fiber f ∗ OP1 (1) of the natural projection f : X → P1 . Recall that X max{i1 a1 + . . . + in an + b + 1, 0} h0 (X, OX (aH + bF )) = i1 +...+in =a,ij ≥0
and apply Theorem 6.3 with D = bF and D = H − bF , with b ≥ 0. In order to get Example 1.6, we are going to apply Corollary 6.5. We fix b := h + 1, so the first inequality is satisfied. Next, just noticing that by definition the (0, h)-defectivity is exactly the h-defectivity, we substitute the prescribed numerical values in the second inequality and after elementary computations we obtain the condition h ≥ 4.
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[33] A. Terracini: “Sulle Vk per cui la variet`a degli Sh (h+1)-seganti ha dimensione minore dell’ordinario“, Rend. Circ. Mat. Palermo, Vol. 31, (1911), pp. 392–396. [34] A. Terracini: “Sulla rappresentazione delle coppie di forme ternarie mediante somme di potenze di forme lineari“, Ann. di Matem. pura ed appl., Vol. 24(3), (1915), pp. 91–100. [35] A. Terracini: “Su due problemi, concernenti la determinazione di alcune classi di superficie, considerati da G. Scorza e F. Palatini“, Atti Soc. Natur. e Matem. Modena, Vol. 5(6), (1921-22), pp. 3–16.
CEJM 2(4) 2004 561–572
Local properties of the solution set of the operator equation in Banach spaces in a neighbourhood of a bifurcation point Joanna Janczewska∗ Department of Technical Physics and Applied Mathematics, Gda´ nsk University of Technology, Narutowicza 11/12, 80–952 Gda´ nsk, Poland
Received 13 January 2004; accepted 7 September 2004 Abstract: In this work we study the problem of the existence of bifurcation in the solution set of the equation F (x, λ) = 0, where F : X × Rk → Y is a C 2 -smooth operator, X and Y are Banach spaces such that X ⊂ Y . Moreover, there is given a scalar product h·, ·i : Y × Y → R 1 that is continuous with respect to the norms in X and Y . We show that under some conditions there is bifurcation at a point (0, λ0 ) ∈ X × Rk and we describe the solution set of the studied equation in a small neighbourhood of this point. c Central European Science Journals. All rights reserved. ° Keywords: bifurcation, finite-dimensional reduction, Fredholm operator, implicit function, variational gradient MSC (2000): 34K18
1
Introduction
Let X and Y be real Banach spaces and F : X × Rk → Y be a continuous map. Suppose that the equation F (x, λ) = 0, (1) where x ∈ X and λ = (λ1 , λ2 , . . . , λk ) ∈ Rk , possesses the trivial family of solutions Λ = {(0, λ) ∈ X × Rk : λ ∈ Rk }. A point (x, λ) such that F (x, λ) = 0 and x 6= 0 is called a nontrivial solution of (1). Bifurcation theory is concerned in part with the existence of nontrivial solutions of (1) ∗
E-mail: [email protected]
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in a small neighbourhood of Λ. A point (0, λ0 ) ∈ Λ is called a bifurcation point of (1) if every neighbourhood of (0, λ0 ) contains a nontrivial solution of (1). Methods of bifurcation theory are often applied in mathematical physics. Let us mention some applications to mechanics of elastic constructions and hydromechanics. In [3] the buckling of a thin elastic plate subject to arbitrary forces and stresses along its boundary is studied by the use of a perturbation theory and a variational method. In [6] to describe a deformation of the minimal interface of two fluids in a vertical tube in a gravitational field one applies a method based on the Crandall-Rabinowitz bifurcation theorem and representation theory. In [9] the buckling of a thin elastic rectangular plate simply supported on sides is studied numerically. In [14] the forms of equilibrium of a thin elastic circular plate lying on an elastic foundation and simply supported along its boundary are investigated via a finite-dimensional reduction and the Krasnosielski bifurcation theorem. Finally, in [16] the buckling of a homogeneous finite beam clamped at the edges to an elastic foundation is studied by a method of a key function due to Sapronov. Assume that F is C 1 -smooth. For every λ ∈ Rk , let Fx0 (0, λ) : X → Y denote the Fr´echet derivative of F with respect to x at (0, λ). Let N (λ) = ker Fx0 (0, λ) and R(λ) = im Fx0 (0, λ). It is easily seen that if Fx0 (0, λ0 ) : X → Y is a Fredholm operator of index zero then a necessary condition for (0, λ0 ) to be a bifurcation point of (1) is dim N (λ0 ) > 0. In this paper we investigate bifurcation at (0, λ0 ) when X is a linear subspace of Y , there is given a scalar product h·, ·i : Y × Y → R1 that is continuous with respect to the norms in X and Y , and F is a C p -smooth map (p ≥ 2) that satisfies the following conditions: (I1 ) F (0, λ) = 0 for every λ ∈ Rk , (I2 ) dim N (λ0 ) = 1, (I3 ) N (λ0 ) ⊥ R(λ0 ), (I4 ) Fx0 (0, λ0 ) : X → Y is a Fredholm operator of index 0. Our aim is to prove a theorem on bifurcation at (0, λ0 ) and a local structure of a solution set of equation (1) in a neighbourhood of a bifurcation point. We apply a kind of finite-dimensional reduction of Liapunov-Schmidt type and the implicit function theorem. We are motivated by applications in mathematical physics [6], [14], [16] in which the problems under considerations (see above) are described by (1) with F that satisfies (I1 )–(I4 ) and is a variational gradient. The main results of this work are Theorem 3.7 and its variational version: Conclusion 3.10. Theorem 3.7 is an analogue of the CrandallRabinowitz bifurcation theorem (see [17], [21]). However, our theorem is formulated in terms of a finite-dimensional reduction and in a variational case it seems to be easier to apply. Conclusion 3.10 is well adapted to a class of nonlinear problems of elasticity described by the von K´arm´an equations with one or a few parameters (see [4], [15], [16]) in the case when the linearization space is one-dimensional. An example is given in Section 4.
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The paper is divided into four sections. In Section 2 we introduce some notions and we briefly sketch a scheme of finite-dimensional reduction. Section 3 is devoted to the study of bifurcation and local properties of the solution set of (1) near a bifurcation point. In Section 4 some applications of our results are indicated. In practice it suffices to suppose that F is defined in a neighbourhood of (0, λ0 ) in X × Rk , but we want to omit inessential details.
2
Finite-dimensional reduction
In this section we describe a kind of a finite-dimensional reduction of the LiapunovSchmidt type. The scheme we present is adapted from [21] ( see also [10], [11], [17], [20]). From now on we assume that X ⊂ Y are real Banach spaces with a scalar product h·, ·i : Y × Y → R1 that is continuous with respect to the norms in X and Y . The norms in X and Y can be defined independently of the scalar product h·, ·i, and the norm in X does not have to be induced by the norm in Y . In particular, X and Y with h·, ·i may be Hilbert spaces. Let F : X × Rk → Y be a C p -smooth map, where p ≥ 1, satisfying conditions: (I1 ), (I3 ), (I4 ) and (I20 ) dim N (λ0 ) = n 6= 0. The aim is to show that under the above assumptions the problem of bifurcation for equation (1) at the point (0, λ0 ) ∈ X × Rk is reducible to the problem of bifurcation for the equation ϕ(ξ, λ) = 0 with a certain map ϕ : S ⊂ R n × Rk → Rn at the point (0, λ0 ) ∈ Rn × Rk . The reader may find the proofs of the propositions given below in [13] and [15]. Proposition 2.1. For every λ ∈ Rk the following equality holds: Y = R(λ) ⊕ N (λ).
(2)
Let G : X × Rn × Rk → Y be a map defined by G(x, ξ, λ) = F (x, λ) +
n X
(ξi − hx, ei i)ei ,
(3)
i=1
where ξ = (ξ1 , ξ2 , . . . , ξn ) and {e1 , e2 , . . . , en } is a fixed orthonormal base of N (λ0 ). Proposition 2.2. The operator G0x (0, 0, λ0 ) : X → Y is an isomorphism. It is easily seen that G is C p -smooth. From the implicit function theorem it follows that there exist two open sets U ⊂ X and S ⊂ Rn × Rk such that 0 ∈ U , (0, λ0 ) ∈ S and the solution set of the equation G(x, ξ, λ) = 0 (4) in U × S is a graph of a certain C p -smooth function x : S → U such that x(0, λ0 ) = 0. Moreover, it is obvious that x(0, λ) = 0 for all (0, λ) ∈ S, because G(0, 0, λ) = 0. Let
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ϕ = (ϕ1 , ϕ2 , . . . , ϕn ) : S → Rn be defined by coordinates as follows: ϕi (ξ, λ) = ξi − hx(ξ, λ), ei i, i = 1, . . . , n.
(5)
Proposition 2.3. (0, λ0 ) ∈ Λ is a bifurcation point of equation (1) if and only if (0, λ0 ) ∈ S is a bifurcation point of equation ϕ(ξ, λ) = 0.
3
(6)
Theorem on bifurcation
In this section our main results are stated and proved. Let F : X × Rk → Y be a C p -smooth map, p ≥ 2, satisfying conditions (I1 )-(I4 ) (see p. 562). Fix e ∈ N (λ0 ) such that he, ei = 1 and denote λ0 = (λ01 , λ02 , . . . , λ0k ). We will describe the solution set of (1) in terms of the finite-dimensional reduction. Notice that now in the formulas of maps G and ϕ there are n = 1 and e1 = e. Differentiating the equality G(x(ξ, λ), ξ, λ) = 0 with respect to ξ at (0, λ0 ) we obtain Fx0 (0, λ0 )x0ξ (0, λ0 ) + (1 − hx0ξ (0, λ0 ), ei)e = 0. From this and (I3 ) it follows that x0ξ (0, λ0 ) = e. Theorem 3.1. There exist open sets V0 ⊂ X and V ⊂ Rk such that (0, λ0 ) ∈ V0 × V and for every (x, λ) ∈ V0 × V we have F (x, λ) = 0 if and only if (hx, ei, λ) ∈ S and x = x(hx, ei, λ). Proof 3.2. Suppose contrary to our claim, that there are no open sets V0 ⊂ X and V ⊂ Rk with the above properties. Then for every n ∈ N there exists (xn , λn ) ∈ X × Rk such that ||xn ||X ≤ n1 , |λn − λ0 | ≤ n1 and one of the following conditions is satisfied: 1. F (xn , λn ) = 0 and (hxn , ei, λn ) ∈ / S, 2. F (xn , λn ) = 0, (hxn , ei, λn ) ∈ S and xn 6= x(hxn , ei, λn ), 3. F (xn , λn ) 6= 0, (hxn , ei, λn ) ∈ S and xn = x(hxn , ei, λn ). If (hxn , ei, λn ) ∈ S and xn = x(hxn , ei, λn ) then F (xn , λn ) = F (x(hxn , ei, λn ), λn ) + (hxn , ei − hx(hxn , ei, λn ), ei)e = G(x(hxn , ei, λn ), hxn , ei, λn ) = 0. Since xn → 0 in X, there exists n0 ∈ N such that xn ∈ U for every n ≥ n0 . If for some n ≥ n0 we have F (xn , λn ) = 0 and (hxn , ei, λn ) ∈ S then 0 = F (xn , λn ) + (hxn , ei − hxn , ei)e = G(xn , hxn , ei, λn ), and so xn = x(hxn , ei, λn ). Since (hxn , ei, λn ) → (0, λ0 ) ∈ S there exists n1 ∈ N such that (hxn , ei, λn ) ∈ S for every n ≥ n1 — a contradiction. The equality hG(x(ξ, λ), ξ, λ), ei = 0 implies ϕ(ξ, λ) = −hF (x(ξ, λ), λ), ei. From (7) we obtain ϕ0ξ (ξ, λ) = −hFx0 (x(ξ, λ), λ)x0ξ (ξ, λ), ei,
(7)
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and hence ϕ0ξ (0, λ0 ) = 0. Moreover, since ϕ(0, λ) = 0 for every (0, λ) ∈ S we have (m) ϕλi λi ...λim (0, λ0 ) = 0 for all i1 , i2 , . . . , im ∈ {1, 2, . . . , k} and m ∈ N . In order to get our 1 2 main result we have to assume that there is i ∈ {1, 2, . . . , k} such that ϕ00ξλi (0, λ0 ) 6= 0. There is no loss of generality if we assume (I5 ) ϕ00ξλk (0, λ0 ) 6= 0. From now on, if λ = (λ1 , λ2 , . . . , λk−1 , λk ) ∈ Rk , λ0 = (λ1 , λ2 , . . . , λk−1 ) ∈ Rk−1 we will write λ = (λ0 , λk ). Proposition 3.3. There exist open sets Ω0 ⊂ R1 × Rk−1 and Ω ⊂ R1 such that (0, λ00 ) ∈ Ω0 , λ0k ∈ Ω and there exists a C p -smooth map f : Ω0 → Ω that satisfies the following conditions: (1) f (0, λ00 ) = λ0k , (2) for every (ξ, λ0 ) ∈ Ω0 and λk ∈ Ω we have ϕ(ξ, λ0 , λk ) = 0 if and only if ξ = 0 or λk = f (ξ, λ0 ) . Proof 3.4. Let ψ : S → R1 be a function defined by Z 1 ψ(ξ, λ) = ϕ0ξ (tξ, λ)dt.
(8)
0
Observe that we have ϕ(ξ, λ) = ξψ(ξ, λ).
(9)
Hence ϕ(ξ, λ) = 0 only if ξ = 0 or ψ(ξ, λ) = 0. From (8) and (I5 ) it follows that ψ(0, λ0 ) = ϕ0ξ (0, λ0 ) = 0 and ψλ0 k (0, λ0 ) = ϕ00ξλk (0, λ0 ) 6= 0. Applying the implicit function theorem we get the desired claim. Let Br (λ00 ) denote a ball in Rk−1 of radius r centered at λ00 , and Bδ (0) a ball in X of radius δ centered at 0. Theorem 3.5. Let f : Ω0 → Ω be a function of Proposition 3.3 and r > 0 be a number such that (−r, r) × Br (λ00 ) ⊂ Ω0 . There exist open sets V˜0 ⊂ X and V˜ ⊂ Br (λ00 ) × Ω such that (0, λ0 ) ∈ V˜0 × V˜ and for every (x, λ) ∈ V˜0 × V˜ we have F (x, λ) = 0 if and only if x = 0 or there exists ξ ∈ (−r, r) such that λk = f (ξ, λ0 ) and x = x(ξ, λ0 , f (ξ, λ0 )). Proof 3.6. There exists δ ∈ (0, r) such that for every x ∈ X if ||x||X < δ then |hx, ei| < r. Let V˜0 = V0 ∩ Bδ (0) and V˜ = V ∩ (Br (λ00 ) × Ω), where V0 ⊂ X and V ⊂ Rk are open sets of Theorem 3.1. Take (x, λ) ∈ V˜0 × V˜ . (⇒) By Theorem 3.1, if F (x, λ) = 0 then (hx, ei, λ) ∈ S and x = x(hx, ei, λ), which gives ϕ(hx, ei, λ) = 0. From Proposition 3.3 it follows that hx, ei = 0 or λ k = f (hx, ei, λ0 ). If hx, ei = 0 then x = x(0, λ) = 0. If λk = f (hx, ei, λ0 ) then x = x(hx, ei, λ0 , f (hx, ei, λ0 )). (⇐) Assume now that x = 0 or there exists ξ ∈ (−r, r) such that λk = f (ξ, λ0 ) and x = x(ξ, λ0 , f (ξ, λ0 )). In the first case, F (x, λ) = F (0, λ) = 0. In the second case, by Proposition 3.3, we have ϕ(ξ, λ) = 0, and hence F (x, λ) = F (x, λ) + ϕ(ξ, λ)e = F (x, λ) + (ξ − hx(ξ, λ), ei)e = F (x, λ) + (ξ − hx, ei)e = G(x, ξ, λ) = 0.
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We are now in a position to prove our main result. Theorem 3.7. Under assumptions (I1 )-(I5 ), the solution set of equation (1) in a certain neighbourhood of (0, λ0 ) ∈ Λ is the union of two sets: Λ and Ξ. The set Ξ is given by Ξ = {(ˆ x(ξ, λ0 ), λ0 , f (ξ, λ0 )) : |ξ| < r, |λ0 − λ00 | < r}, where xˆ and f are C p -smooth functions such that xˆ(0, λ00 ) = 0, f (0, λ00 ) = λ0k , xˆ0ξ (0, λ00 ) = ϕ00
ϕ00 (0,λ0 )
(0,λ0 )
e, fξ0 (0, λ00 ) = − 12 ϕ00ξξ (0,λ0 ) , xˆ0λs (0, λ00 ) = 0 and fλ0 s (0, λ00 ) = − ϕ00ξλs (0,λ0 ) for every s ∈ ξλk
ξλk
{1, 2, . . . , k − 1}. Moreover, the intersection of Λ and Ξ in a sufficiently small neighbourhood of (0, λ 0 ) can be parametrized as follows ˆ 0 ), λ0 )) : |λ0 − λ0 | < %} IΛ,Ξ = {(0, λ0 , f (ξ(λ 0 ˆ 0 ) = 0 and ξˆ0 (λ0 ) = 0 for where 0 < % ≤ r and ξˆ is a C p -smooth function such that ξ(λ 0 0 λs every s ∈ {1, 2, . . . , k − 1}, which gives that (0, λ0 ) is a bifurcation point of (1). Proof 3.8. Let f : Ω0 → Ω be a function of Proposition 3.3. Fix r > 0 such that (−r, r)× Br (λ00 ) ⊂ Ω0 . Let xˆ : (−r, r) × Br (λ00 ) → X be given by xˆ(ξ, λ0 ) = x(ξ, λ0 , f (ξ, λ0 )). Then f (0, λ00 ) = λ0k and xˆ(0, λ00 ) = x(0, λ0 ) = 0. Differentiating xˆ we get xˆ0ξ (0, λ00 ) = e and xˆ0λs (0, λ00 ) = 0 for every s ∈ {1, 2, . . . , k − 1}. Moreover, differentiating the equality ψ 0 (0,λ0 )
ϕ00 (0,λ0 )
ψ(ξ, λ0 , f (ξ, λ0 )) = 0 we obtain fξ0 (0, λ00 ) = − ψ0ξ (0,λ0 ) = − 21 ϕ00ξξ (0,λ0 ) and fλ0 s (0, λ00 ) = ψ 0 (0,λ0 ) − ψλ0 s (0,λ0 ) λ k
=
ϕ00 (0,λ0 ) − ϕ00ξλs (0,λ0 ) ξλ
λk
ξλk
for every s ∈ {1, 2, . . . , k − 1}. From Theorem 3.5 it follows
k
that there exist open sets V˜0 ⊂ X and V˜ ⊂ Br (λ00 ) × Ω such that (0, λ0 ) ∈ V˜0 × V˜ and {(x, λ) ∈ V˜0 × V˜ : F (x, λ) = 0} = {(x, λ) ∈ V˜0 × V˜ : x = 0}∪{(x, λ) ∈ V˜0 × V˜ : ∃ξ∈(−r,r) x = x(ξ, λ0 , f (ξ, λ0 )) ∧ λk = f (ξ, λ0 )} = (Λ ∪ Ξ) ∩ V˜0 × V˜ . A point (x, λ) ∈ Λ ∩ Ξ only if it satisfies the following system 0 x = xˆ(ξ, λ ), λk = f (ξ, λ0 ), ξ ∈ (−r, r), λ0 ∈ Br (λ00 ), x = 0.
Since xˆ(0, λ00 ) = 0 and xˆ0ξ (0, λ00 ) = e 6= 0, there exist: 0 < % ≤ r, an open set B ⊂ (−r, r) ˆ 0 ) = 0 and such that 0 ∈ B and a C p -smooth function ξˆ: B% (λ00 ) → B such that ξ(λ 0 ˆ 0 ). Differentiating for all (ξ, λ0 ) ∈ B × B% (λ00 ) we have xˆ(ξ, λ0 ) = 0 only if ξ = ξ(λ ˆ 0 ), λ0 ) = 0 for every ˆ 0 ), λ0 ) = 0 we receive xˆ0 (ξ(λ ˆ 0 ), λ0 )ξˆ0 (λ0 ) + xˆ0 (ξ(λ the equality xˆ(ξ(λ λs ξ λs s ∈ {1, 2, . . . , k − 1}, and hence ξˆ0 (λ0 ) = 0. Summarizing IΛ,Ξ ⊂ Λ ∩ Ξ and in a λs
0
sufficiently small neighbourhood of (0, λ0 ) the intersection Λ ∩ Ξ is equal to IΛ,Ξ . Conclusion 3.9. Assume that (I1 )-(I5 ) hold and k = 2. Then the solution set of (1) in a small neighbourhood of (0, λ0 ) ∈ Λ is the union of two surfaces: Λ and Ξ. The surface Ξ can be parametrized as follows Ξ = {(ˆ x(ξ, λ1 ), λ1 , f (ξ, λ1 )) : (ξ, λ1 ) ∈ (−r, r) × (λ01 − r, λ01 + r)},
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where xˆ : (−r, r) × (λ01 − r, λ01 + r) → X and f : (−r, r) × (λ01 − r, λ01 + r) → R1 are C p smooth functions such that xˆ(0, λ01 ) = 0, f (0, λ01 ) = λ02 , xˆ0ξ (0, λ01 ) = e, xˆ0λ1 (0, λ01 ) = 0, ϕ00
ϕ00 (0,λ01 )
(0,λ01 )
1 fξ0 (0, λ01 ) = − 12 ϕ00ξξ (0,λ01 ) and fλ0 1 (0, λ01 ) = − ϕξλ 00 (0,λ ) . In a sufficiently small neighbour01 ξλ2
ξλ2
hood of (0, λ0 ) the surfaces Λ and Ξ intersect only along the curve ˆ 1 ), λ1 )) : λ1 ∈ (λ01 − %, λ01 + %)}, IΛ,Ξ = {(0, λ1 , f (ξ(λ where 0 < % ≤ r and ξˆ: (λ01 − %, λ01 + %) → (−r, r) is a C p -smooth function such that ˆ 01 ) = ξˆ0 (λ01 ) = 0, and hence (0, λ0 ) is a bifurcation point of (1). ξ(λ Let us consider the following condition: (I30 ) F : X × Rk → Y is a variational gradient of a certain functional E : X × R k → R1 with respect to the scalar product h·, ·i, i.e. for all x, y ∈ X and λ ∈ R k Ex0 (x, λ)y = hF (x, λ), yi. It is evident that (I30 ) implies (I3 ). Furthermore, by formula (7) we obtain (3)
ϕ00ξλs (0, λ0 ) = −Exxλs (0, λ0 )(e, e, 1)
(10)
for s ∈ {1, 2, . . . , k}. From this it follows that if F satisfies (I 30 ) then (I5 ) can be replaced by the equivalent condition: (3) (I50 ) Exxλk (0, λ0 )(e, e, 1) 6= 0. By (7) we also obtain (3) ϕ00ξξ (0, λ0 ) = −Exxx (0, λ0 )(e, e, e). (11) Summarizing, in a variational case we have the following result. Conclusion 3.10. Under assumptions: (I1 ), (I2 ), (I30 ), (I4 ) and (I50 ), the solution set of equation (1) in a certain neighbourhood of (0, λ0 ) ∈ Λ is the union of two sets: Λ and Ξ. The set Ξ is given by Ξ = {(ˆ x(ξ, λ0 ), λ0 , f (ξ, λ0 )) : |ξ| < r, |λ0 − λ00 | < r}, where xˆ and f are C p -smooth functions such that xˆ(0, λ00 ) = 0, f (0, λ00 ) = λ0k , xˆ0ξ (0, λ00 ) = e, fξ0 (0, λ00 ) = − 21
(3)
Exxx (0,λ0 )(e,e,e) (3) Exxλ (0,λ0 )(e,e,1) k
(3)
, xˆ0λs (0, λ00 ) = 0 and fλ0 s (0, λ00 ) = −
Exxλs (0,λ0 )(e,e,1) (3) Exxλ (0,λ0 )(e,e,1) k
for every
s ∈ {1, 2, . . . , k − 1}. Moreover, the intersection of Λ and Ξ in a sufficiently small neighbourhood of (0, λ 0 ) can be parametrized as follows ˆ 0 ), λ0 )) : |λ0 − λ0 | < %} IΛ,Ξ = {(0, λ0 , f (ξ(λ 0 ˆ 0 ) = 0 and ξˆ0 (λ0 ) = 0 for where 0 < % ≤ r and ξˆ is a C p -smooth function such that ξ(λ 0 0 λs every s ∈ {1, 2, . . . , k − 1}, which gives that (0, λ0 ) is a bifurcation point of (1).
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Applications
It is obvious that if we assume that F is a map from a small neighbourhood of the point (0, λ0 ) in X × Rk to Y , our results remain true. After this remark we are ready to give an example of application of Conclusion 3.10 to mathematical physics. All the results of Section 4 were proved either in [12] or [15]. However, to make this exposition self-sufficient we give the main ideas of the proofs. ¯ denote the real H¨older space of functions For every m ∈ N and µ ∈ (0, 1), let C m,µ (D) defined on D = {(u, v) ∈ R2 : u2 + v 2 < 1} with the standard norm ¯ = max sup {|Dα x(u, v)| : (u, v) ∈ D} + ||x; C m,µ (D)|| |α|≤m ½ α ¾ |D x(u, v) − D α x(¯ u, v¯)| max sup : (u, v), (¯ u, v¯) ∈ D, (u, v) 6= (¯ u, v¯) , |α|≤m |(u − u¯, v − v¯)|µ |α|
where Dα x = ∂ α∂1 u∂xα2 v , α = (α1 , α2 ) ∈ N0 × N0 , N0 = N ∪ {0} and |α| = α1 + α2 . It is ¯ is a Banach space (see [1]). Let well-known that C m,µ (D) • • • •
4,µ ¯ ¯ : ∆f |∂D = f |∂D = 0}, C0,0 (D) = {f ∈ C 4,µ (D) 2,µ ¯ ¯ : f |∂D = 0}, C0 (D) = {f ∈ C 2,µ (D) 4,µ ¯ 4,µ ¯ X = C0,0 (D) × C0,0 (D), ¯ × C 0,µ (D). ¯ Y = C 0,µ (D)
The norms in X and Y are defined by coordinates. That is as the maximum (or the sum) of norms of both coordinates of a given element. The function given by ZZ 1 h(x1 , x2 ), (y1 , y2 )i = (x1 y1 + x2 y2 )dudv π D is a scalar product in Y , which is continuous with respect to the norms in X and Y . We 2 define F : X × R+ → Y as follows 1 F (x, λ) = (∆2 x1 − [x1 , x2 ] + 2λ1 ∆x1 + λ2 x1 − γx31 , −∆2 x2 − [x1 , x1 ]), 2
(12)
where R+ = (0, +∞), x = (x1 , x2 ), λ = (λ1 , λ2 ), γ is a positive constant and [·, ·] : X → Y is given by ∂ 2 x1 ∂ 2 x2 ∂ 2 x1 ∂ 2 x2 ∂ 2 x1 ∂ 2 x2 [x1 , x2 ] = − 2 + . ∂u2 ∂v 2 ∂u∂v ∂u∂v ∂v 2 ∂u2 The equation F (x, λ) = 0 (13) with F given by (12) is called the von K´arman equation for a thin circular elastic plate which lies on an elastic base and is uniformly radially compressed along its boundary. In mechanics x1 is a deflection function, x2 is a stress function, λ1 is a value of a compressing force, λ2 and γ are parameters of an elastic foundation. The solutions of (13) lying in a sufficiently small neighbourhood of the set of trivial solutions of (13) are called the forms
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of equilibrium of a plate. The map F is C ∞ -smooth and an easy computation shows that for all y = (y1 , y2 ) ∈ X Fx0 (x, λ)y = (∆2 y1 − [y1 , x2 ] − [x1 , y2 ] + 2λ1 ∆y1 + λ2 y1 − 3γx21 y1 , −∆2 y2 − [x1 , y1 ]). (14) 2 Let E : X × R+ → R1 be given by ZZ ¡ ¢ 1 E(x, λ) = (∆x1 )2 − (∆x2 )2 − [x1 , x1 ]x2 dudv + 2π D ! õ ¶2 µ ¶2 ! ZZ à 1 1 ∂x1 ∂x1 −2λ1 + λ2 x21 − γx41 dudv. + 2π ∂u ∂v 2 D
(15)
E is easily seen to be C ∞ -smooth. Theorem 4.1 (see Th. 2.4 of [12]). The map F is a variational gradient of the functional E with respect to the scalar product h·, ·i. 2 Sketch of the proof 4.2. For all x, y ∈ X and λ ∈ R+ , we have ZZ ZZ d 1 1 0 Ex (x, λ)y = E(x + ty, λ)|t=0 = ∆x1 ∆y1 dudv − ∆x2 ∆y2 dudv dt π ZDZ π D ZZ 1 1 − [x1 , y1 ]x2 dudv − [x1 , x1 ]y2 dudv π 2π D D ¶ µ ZZ 1 ∂x1 ∂y1 ∂x1 ∂y1 − 2λ1 dudv + π ∂u ∂u ∂v ∂v Z ZD 1 (λ2 x1 y1 − γx31 y1 )dudv. + π D
Integrating by part we receive ZZ ZZ ∆x1 ∆y1 dudv = (∆2 x1 )y1 dudv, Z ZD Z ZD ∆x2 ∆y2 dudv = (∆2 x2 )y2 dudv, Z ZD Z ZD [x1 , y1 ]x2 dudv = [x1 , x2 ]y1 dudv D
and
ZZ µ D
∂x1 ∂y1 ∂x1 ∂y1 + ∂u ∂u ∂v ∂v
D
¶
dudv = −
ZZ
(∆x1 )y1 dudv. D
Hence Ex0 (x, λ)y = hF (x, λ), yi, which completes the proof. 2 Theorem 4.3 (see Th. 2.2 of [12]). For every λ ∈ R+ , Fx0 (0, λ) : X → Y is a Fredholm map of index 0. 2 Sketch of the proof 4.4. Fix λ ∈ R+ . By (14) we get
Fx0 (0, λ)y = (∆2 y1 + 2λ1 ∆y1 + λ2 y1 , −∆2 y2 ).
(16)
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We can write (16) as Fx0 (0, λ)y = A(y) + B(y), where A, B : X → Y are given as follows: A(y) = (∆2 y1 , −∆2 y2 ),
B(y) = (2λ1 ∆y1 + λ2 y1 , 0).
It is known that ∆ : C02,µ (D) → C 0,µ (D) is an isomorphism. Moreover, it is a simple matter to check that B is compact, which finishes the proof. Let Jk : R → R, k ∈ N0 , denote the k-th Bessel function. It is well-known (see [8], [18]) that α ∈ R is an eigenvalue of ∆ : C02,µ (D) → C 0,µ (D) if and only if α < 0 and there √ √ is k ∈ N0 such that Jk ( −α) = 0. Furthermore, if J0 ( −α) = 0 then the eigenspace √ corresponding to α is one-dimensional. If Jk ( −α) = 0 for a certain k ∈ N then the corresponding eigenspace is two-dimensional. √ √ 2 , let δ = (λ1 )2 −λ2 , a = −λ1 − δ and b = −λ1 + δ. Of course, For λ = (λ1 , λ2 ) ∈ R+ 4,µ a and b are determined on condition δ ≥ 0. Let ∆2 + 2λ1 ∆ + λ2 I : C0,0 (D) → C 0,µ (D) and ∆ − aI, ∆ − bI : C02,µ (D) → C 0,µ (D), where I(h) = h are natural embeddings of the appropriate H¨older spaces. Lemma 4.5 (see Lemmas 4.1-4.3 of [12]). Under the above assumptions: (i) If δ < 0 then ker(∆2 + 2λ1 ∆ + λ2 I) = {0}. (ii) If δ = 0 then ker(∆2 + 2λ1 ∆ + λ2 I) = ker(∆ + λ1 I). (iii) If δ > 0 then ker(∆2 + 2λ1 ∆ + λ2 I) = ker(∆ − aI) ⊕ ker(∆ − bI). By (16), N (λ) = ker(∆2 + 2λ1 ∆ + λ2 I) × {0}. From this and Lemma 4.5 we obtain what follows. Theorem 4.6. dim N (λ) = 1 if and only if one of the below conditions is satisfied: √ (I) δ = 0 and J0 ( λ1 ) = 0, √ √ (II) δ > 0, J0 ( −a) = 0 and Jk ( −b) 6= 0 for every k ∈ N0 , √ √ (III) δ > 0, J0 ( −b) = 0 and Jk ( −a) 6= 0 for every k ∈ N0 . Suppose that λ0 = (λ01 , λ02 ) and dim N (λ0 ) = 1. Fix e = (e1 , 0) ∈ N (λ0 ) such that he, ei = 1. Set ( a0 if (I) or (II), c0 = b0 if (III), √ √ where a0 = −λ01 − δ0 , b0 = −λ01 + δ0 and δ0 = (λ01 )2 − λ02 . A trivial verification combining Theorem 4.1 with (14) shows that ZZ 2 000 (∆y1 )z1 dudv, Exxλ1 (x, λ)(y, z, 1) = π Z ZD 1 000 Exxλ (x, λ)(y, z, 1) = y1 z1 dudv, 2 π D
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and 000 Exxx (x, λ)(y, z, w)
ZZ 1 =− ([y1 , z2 ] + [y2 , z1 ] + 6γx1 y1 z1 )w1 dudv π Z ZD 1 − [y1 , z1 ]w2 dudv, π D
where x = (x1 , x2 ) , y = (y1 , y2 ), z = (z1 , z2 ), w = (w1 , w2 ). From this and Lemma 4.5 we receive ZZ 2 000 Exxλ1 (0, λ0 )(e, e, 1) = (∆e1 )e1 dudv = 2c0 he, ei = 2c0 , π Z ZD 1 000 Exxλ (0, λ0 )(e, e, 1) = e21 dudv = he, ei = 1, 2 π D 000 Exxx (0, λ0 )(e, e, e) = 0. Applying Conclusion 3.10 we get the following theorem. 2 satisfy the above assumptions. Then the solution set of Theorem 4.7. Let λ0 ∈ R+ 2 equation (13) in a certain neighbourhood of (0, λ0 ) ∈ X × R+ is the union of two sets: Λ and Ξ. The set Ξ is given by
Ξ = {(ˆ x(ξ, λ1 ), λ1 , f (ξ, λ1 )) : |ξ| < r, |λ1 − λ01 | < r}, where xˆ and f are C ∞ -smooth functions such that xˆ(0, λ01 ) = 0, f (0, λ01 ) = λ02 , xˆ0ξ (0, λ01 ) = e, fξ0 (0, λ01 ) = 0, xˆ0λ1 (0, λ01 ) = 0 and fλ0 1 (0, λ01 ) = −2c0 . Moreover, the intersection of Λ and Ξ in a sufficiently small neighbourhood of (0, λ 0 ) can be parametrized as follows ˆ 1 ), λ1 )) : |λ1 − λ01 | < %}, IΛ,Ξ = {(0, λ1 , f (ξ(λ ˆ 01 ) = 0 and ξˆ0 (λ01 ) = 0, where 0 < % ≤ r and ξˆ is a C ∞ -smooth function such that ξ(λ which gives that (0, λ0 ) is a bifurcation point of (13).
References [1] R.A. Adams: Sobolev Spaces, Acad. Press, New York, 1975. [2] S.S. Antman: Nonlinear Problems of Elasticity, Springer-Verlag, Appl. Math. Sci. 107, Berlin, 1995. [3] M.S. Berger: “On von K´arman’s equations and the buckling of a thin elastic plate, I. The clamped plate”, Comunications on Pure and Applied Mathematics, Vol. 20, (1967), pp. 687–719. [4] F. Bloom, D. Coffin: Handbook of Thin Plate Buckling and Postbuckling, Chapman and Hall/CRC, Boca Raton, 2001. [5] A.Yu. Borisovich: “Bifurcation of a capillary minimal surface in a weak gravitational field”, Sbornik: Mathematics, Vol. 188, (1997), pp. 341–370.
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[6] A.Yu. Borisovich, W. Marzantowicz: “Bifurcation of the equivariant minimal interfaces in a hydromechanics problem”, Abstract and Applied Analysis, Vol. 1, (1996), pp. 291–304. [7] A.Yu. Borisovich , Yu. Morozov , Cz. Szymczak: Bifurcation of the forms of equilibrium of nonlinear elastic beam lying on the elastic base, Preprint 136, the University of Gda´ nsk, 2000. [8] J.W. Brown, R.V. Churchill: Fourier Series and Boundary Value Problems, McGrawHill Companies, New York, 2001. [9] M.S. Chen, C.S. Chien, “Multiple bifurcation in the von K´arm´an equations”, SIAM J. Sci. Comput., Vol. 6, (1997), pp. 1737–1766. [10] S.N. Chow, J.K. Hale, Methods of Bifurcation Theory, Springer–Verlag, New York, 1982. [11] M. Golubitsky, D.G. Schaeffer: Singularities and Groups in Bifurcation Theory, Springer–Verlag, Applied Mathematical Sciences 51, New York, 1985. [12] J. Janczewska: “Bifurcation in the solution set of the von K´arman equations of an elastic disk lying on an elastic foundation”, Annales Polonici Mathematici, Vol. 77, (2001), pp. 53–68. [13] J. Janczewska: “The necessary and sufficient condition for bifurcation in the von K´arman equations”, Nonlinear Differential Equations and Applications, Vol. 10, (2003), pp. 73–94. [14] J. Janczewska: “Application of topological degree to the study of bifurcation in the von K´arman equations”, Geometriae Dedicata, Vol. 91, (2002), pp. 7–21. [15] J. Janczewska: The study of bifurcation in the von K´arman equations. Applying of topological methods and finite dimensional reductions for operators of Fredholm’s type, Ph.D. Thesis, Department of Mathematics and Physics, the University of Gda´ nsk, 2002. [in Polish] [16] Yu. Morozov: The study of the nonlinear model which describes the equilibrium forms, fundamental frequencies and modes of oscillations of a finite beam on an elastic foundation, Ph.D. Thesis, Department of Applied Mathematics, the University of Voronezh, 1998. [in Russian] [17] L. Nirenberg: Topics in Nonlinear Functional Analysis, Courant Inst. of Math. Sciences, New York, 1974. [18] A.A. Samarski, A.N. Tichonov: Equations of Mathematical Physics, PWN, Warsaw, 1963. [19] Yu.I. Sapronov: “Finite dimensional reductions in smooth extremal problems”, Uspehi Mat. Nauk, Vol. 1, (1996), pp. 101–132. [20] V.A. Trenogin, M.M. Vainberg: Theory of Branching of Solutions of Nonlinear Equations, Nauka, Moscow, 1969. [21] E. Zeidler: Nonlinear Functional Analysis and its Applications, Springer–Verlag, Berlin, 1986.
CEJM 2(4) 2004 573–583
On the existence of solutions for nonlinear impulsive periodic viable problems Tiziana Cardinali1∗ , Raffaella Servadei2† 1
Department of Mathematics and Computer Science, University of Perugia, via Vanvitelli 1, Perugia 06123 Italy 2 Department of Mathematics, University of Rome ‘Tor Vergata’, via della Ricerca Scientifica, Roma 00133 Italy
Received 18 May 2004; accepted 8 August 2004 Abstract: In this paper we prove the existence of periodic solutions for nonlinear impulsive 0 viable problems monitored by differential inclusions of the type x (t) ∈ F (t, x(t)) + G(t, x(t)). Our existence theorems extend, in a broad sense, some propositions proved in [10] and improve a result due to Hristova-Bainov in [13]. c Central European Science Journals. All rights reserved. ° Keywords: impulsive periodic viable differential inclusions and equations, canonical domain, Bouligand contingent cone, lower semicontinuity and upper semicontinuity of set valued maps MSC (2000): 34A37, 34A60, 34B15
1
Introduction
Impulsive differential equations, a new branch of the theory of ordinary differential equations, describe evolution processes which at a certain moment change their state rapidly. In the mathematical simulation of such processes it is convenient to assume that this change takes place momentarily and that the process changes its state by jump. Processes of such character are observed in numerous fields of science and technology: mechanics, population dynamics, theoretical physics, industrial robotics, pharmacokinetics, chemical technology, biotechnology, multiple-phase economic dynamics, stock management in production theory and so on. The qualitative investigation of impulsive differential equations began in 1960 with the work of Mil’man - Myshkis (see [16]). Some monographs related ∗ †
E-mail: [email protected] E-mail: [email protected]
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to this new subject have appeared (see Samoilenko - Perestyuk [17], Lakshmikantham - Bainov - Simeonov [15], Bainov - Simeonov [2, 3], Bainov - Covachev [1]). Recently, the attention has been given to impulsive differential inclusions and interesting results concerning the existence of solutions for Cauchy problems and for periodic problems were obtained (see, for instance, Benchohra - Boucherif [4, 5], Benchohra - Henderson - Ntouyas [7], Hristova - Bainov [13], Watson [18], Benchohra - Henderson - Ntouyas Ouahabi [8, 9], Frigon - O’Regan [12]). In this paper we study the existence of periodic solutions for some impulsive viable problems. In particular, in section 3, we consider the following problem: 0 x (t) ∈ F (t, x(t)) + G(t, x(t)) a.e. t ∈ [0, T ] \ {t1 , . . . , tp } (P) x(t+ for any k ∈ {1, . . . , p} k ) = x(tk ) + Ik (x(tk )) x(0) = x(T ),
where Ω ⊂ RN , N ≥ 1, is a canonical domain, F, G : [0, T ] × Ω → P(RN ), P(RN ) := {S : S ⊆ RN }, are set valued maps, 0 = t0 < t1 < · · · < tp+1 = T , Ik : Ω → RN is an impulse function for k ∈ {1, . . . , p}, and x(t+ ) = lim+ x(s), and we want the solutions of s→t
the problem to remain in the fixed canonical domain Ω. We get existence results for the impulsive viable problem (P) (see Theorem 3.1 and Corollary 3.2) by using a result obtained in [10] (see Theorem 3.2). We note that in [11] the authors have studied the impulsive problem (P) considering the particular situation N = 1 and the not viable case. In this paper we study the viable problem in the N -dimensional case. We point out that in order to obtain that the solution of the impulsive problem (P) satisfies the viability property (i.e. x(t) ∈ Ω for any t ∈ Ω) it is necessary to require the Nagumo-type tangential condition. In section 4 we obtain, as a consequence of Theorem 3.1 and Corollary 3.2, two existence theorems (see Corollaries 4.2 and 4.3) for the impulsive periodic viable problem without the perturbation G: x0 (t) ∈ F (t, x(t)) a.e. t ∈ [0, T ] \ {t1 , . . . , tp } (F) x(t+ k ) = x(tk ) + Ik (x(tk )) for any k ∈ {1, . . . , p} x(0) = x(T ).
These results extend, in a broad sense, Theorems 3.1 and 3.2 of [10] (see Remark 4.6). We observe that, with the hypothesis of lower semicontinuity on F , in literature there are only results about the existence of solutions for impulsive problems without viability (see [8, 9] ). Moreover, we remark that, in the field of single valued maps, from Theorem 3.1 we obtain Theorem 4.7. This proposition improves a result due to Hristova - Bainov (see [13], Theorem 2), because for us the single valued map f is not necessarily continuous on [0, T ] × Ω, but only continuous with respect to the second variable (see Remark 4.8). Moreover, we do not require a Lipschitz condition on f (see Remark 4.9).
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Preliminaries
Let Ω ⊂ RN , N ≥ 1, be a domain with boundary ∂Ω and let us denote by Ω = Ω ∪ ∂Ω its closure and by int Ω the interior of Ω. We say that Ω is a canonical domain if Ω is bounded, convex and there exists a finite family of real valued continuously differentiable © ª maps Φi i∈{1,...,q} such that the following conditions hold q n o \ (Ω1) Ω = x ∈ RN : Φi (x) ≤ 0 ; i=1
(Ω2) if there exist x0 ∈ ∂Ω and i ∈ {1, . . . , q} such that Φi (x0 ) = 0,
then ∇Φi (x0 ) 6= 0 (see [13]). Let us note that a compact interval of R is a canonical domain. © ª Let Ω be a canonical domain defined by the family Φi i∈{1,...,q} and let us define α(x) =
n
o i ∈ {1, . . . , q} : Φi (x) = 0 , for any x ∈ ∂Ω
and PΩ (x) =
n o y ∈ RN : h ∇Φi (x), y i ≤ 0, ∀i ∈ α(x) if x ∈ ∂Ω RN
if x ∈ Ω,
where h ·, · i is the scalar product in RN . We recall the notion of the Bouligand contingent cone to Ω at x ∈ Ω ¾ ½ ρ(x + λy, Ω) N =0 , TΩ (x) = y ∈ R : lim inf λ→0+ λ where ρ(z, Ω) = inf k z − v k. v∈Ω
We note that PΩ (x) = TΩ (x) for any x ∈ Ω (see [18], Remark 1.3). Moreover, TΩ (x) is convex and closed for any x ∈ Ω and 0 ∈ TΩ (x) for any x ∈ Ω. Let F : [0, T ] × Ω → P(RN ) be a set valued map. F is said to be lower semicontinuous (l.s.c.) at (t¯, x¯) ∈ [0, T ] × Ω if, for any open set A ⊆ RN such that A ∩ F (t¯, x¯) 6= ∅, there exists a neighbourhood U of (t¯, x¯) such that A∩F (t, x) 6= ∅ for any (t, x) ∈ U ∩([0, T ]×Ω). F is said to be upper semicontinuous (u.s.c.) at (t¯, x¯) ∈ [0, T ] × Ω if, for any open set A ⊆ RN such that F (t¯, x¯) ⊆ A, there exists a neighbourhood U of (t¯, x¯) such that F (t, x) ⊆ A, for any (t, x) ∈ U ∩ ([0, T ] × Ω). Now we give the definition of solution for the impulsive periodic viable problem (P). Definition 2.1. A solution of problem (P) is a function x : [0, T ] → Ω, absolutely continuous in the closed interval [0, t1 ] and in the interval ]tk , tk+1 ] for any k ∈ {1, . . . , p}, such that
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0
x (t) ∈ F (t, x(t)) + G(t, x(t)) a.e. t ∈ [0, T ] \ {t1 , . . . , tp }; x(t+ k ) = x(tk ) + Ik (x(tk ))
for any k ∈ {1, . . . , p};
and x(0) = x(T ). To obtain our existence results for problem (P) we need the following theorem (see [10], Theorem 3.2): Theorem 2.2. Let Ω ⊂ RN , N ≥ 1, be a canonical domain such that int Ω 6= ∅ and let H : [0, T ] × Ω → P(RN ) be a set valued map such that the following conditions hold (H1) H(t, x) is nonempty, convex, closed a.e. on [0, T ], ∀ x ∈ Ω; (H2) H(t, ·) is u.s.c. in Ω, a.e. on [0, T ]; (H3) ∃ (Hn )n∈N , Hn : [0, T ] × Ω → P(RN ) such that (H3.1) Hn (t, x) is nonempty, convex, closed, ∀ (t, x) ∈ [0, T ] × Ω, ∀ n ∈ N; (H3.2) Hn is l.s.c. in [0, T ] × Ω, ∀ n ∈ N; (H3.3) Hn (t, x) ∩ TΩ (x) 6= ∅, ∀ (t, x) ∈ [0, T ] × Ω, ∀ n ∈ N; (H3.4) ∃ γ ∈ L∞ ([0, T ]) such that k Hn (t, x) k≤ γ(t) a.e. on [0, T ], ∀ x ∈ Ω, ∀ n ∈ N; (H3.5) a.e. t ∈ [0, T ], ∀ ² > 0 ∃ n ¯=n ¯ (², t) ∈ N such that Hn (t, x) ⊆ H(t, x) + B(0, ²), ∀n ≥ n ¯, ∀ x ∈ Ω (where B(0, ²) is the ball with center 0 and radius ²); and let Ik : Ω → RN , k ∈ {1, . . . , p}, be an impulse function such that (I4) Ik is continuous in Ω; (I5) x + Ik (x) ∈ Ω, ∀ x ∈ Ω. Then, there exists a solution of the following impulsive periodic viable problem
0 x (t) ∈ H(t, x(t)) a.e. t ∈ [0, T ] \ {t1 , . . . , tp } (H) x(t+ k ) = x(tk ) + Ik (x(tk )) for any k ∈ {1, . . . , p} x(0) = x(T ).
In [10] we prove Theorem 2.2 using an approximation argument together with a result due to Hristova-Bainov about the existence of a viable periodic solution of impulsive differential equations (see [13]).
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577
Existence results for problem (P)
In this section we prove our existence results for the impulsive periodic viable problem (P). Theorem 3.1. Let Ω ⊂ RN , N ≥ 1, be a canonical domain such that int Ω 6= ∅ and let F, G : [0, T ] × Ω → P(RN ) be set valued maps such that the following conditions hold (F 1) ∃ θ : [0, T ] × Ω → P(RN ) such that (F 1.1) θ is l.s.c. in [0, T ] × Ω; (F 1.2) co θ(t, x) ⊆ F (t, x), ∀ (t, x) ∈ [0, T ] × Ω; (F 1.3) θ(t, x) ⊆ TΩ (x), ∀ (t, x) ∈ [0, T ] × Ω; (G1) G(t, x) is nonempty, convex, compact a.e. on [0, T ], ∀ x ∈ Ω; (G2) G(t, ·) is u.s.c. in Ω, a.e. on [0, T ]; (G3) ∃ (Gn )n∈N , Gn : [0, T ] × Ω → P(RN ) such that (G3.1) Gn (t, x) is nonempty, convex, closed, ∀ (t, x) ∈ [0, T ] × Ω, ∀ n ∈ N; (G3.2) Gn is l.s.c. in [0, T ] × Ω, ∀ n ∈ N; (G3.3) Gn (t, x) ∩ TΩ (x) 6= ∅, ∀ (t, x) ∈ [0, T ] × Ω, ∀ n ∈ N; (G3.4) ∃ γ ∈ L∞ ([0, T ]) such that k Gn (t, x) k≤ γ(t) a.e. on [0, T ], ∀ x ∈ Ω, ∀ n ∈ N; (G3.5) a.e. t ∈ [0, T ], ∀ ² > 0 ∃ n ¯=n ¯ (², t) ∈ N such that Gn (t, x) ⊆ G(t, x) + B(0, ²), ∀n ≥ n ¯ , ∀ x ∈ Ω; and let Ik : Ω → RN , k ∈ {1, . . . , p}, be an impulse function such that (I4) Ik is continuous in Ω; (I5) x + Ik (x) ∈ Ω, ∀ x ∈ Ω. Then, there exists a solution of the impulsive periodic viable problem (P). Proof: In order to prove the existence of a solution for problem (P) we will introduce a suitable impulsive periodic viable problem and we will prove that it has a solution by means of Theorem 2.2. First of all, let us note that the set valued map co θ : [0, T ] × Ω → P(RN ) is l.s.c. in [0, T ] × Ω (see Proposition 2.42 of [14]). Then, by Michael’s Selection Theorem, there exists a continuous selection h : [0, T ] × Ω → RN for the set valued map co θ, that is h(t, x) ∈ co θ(t, x), ∀(t, x) ∈ [0, T ] × Ω.
(1)
Now, let us consider the problem (H) defined on Theorem 2.2 with H(t, x) = {h(t, x)} +
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G(t, x), ∀(t, x) ∈ [0, T ] × Ω and let us verify that the set valued map H satisfies all the hypotheses of Theorem 2.2. By (G1), H(t, x) is nonempty, convex and closed a.e. on [0, T ], ∀ x ∈ Ω. Taking into account (G1), (G2) and the continuity of h, we can deduce that H(t, ·) is u.s.c. in Ω a.e. t ∈ [0, T ] (see Proposition 2.59 of [14]). Finally, let us consider the sequence of set valued maps (Hn )n∈N , where Hn : [0, T ] × Ω → P(RN ) is defined in this way: Hn (t, x) = {h(t, x)} + Gn (t, x), ∀ (t, x) ∈ [0, T ] × Ω, ∀ n ∈ N. By (G3.2) and the continuity of h, we can say that Hn is l.s.c. in [0, T ] × Ω, ∀ n ∈ N (see again Proposition 2.59 of [14]). By (1) and (F 1.3), we have that h(t, x) ∈ co θ(t, x) ⊆ TΩ (x), ∀ (t, x) ∈ [0, T ] × Ω.
(2)
Moreover, by (G3.3), there exists yn,t,x ∈ Gn (t, x) ∩ TΩ (x), ∀ (t, x) ∈ [0, T ] × Ω, ∀ n ∈ N. So, by Propositions 5.7 and 5.32 of [14] and by (2), we have that h(t, x) + yn,t,x ∈ TΩ (x), ∀ (t, x) ∈ [0, T ] × Ω, ∀ n ∈ N. Therefore, by definition of Hn , we have Hn (t, x) ∩ TΩ (x) 6= ∅, ∀ (t, x) ∈ [0, T ] × Ω, ∀ n ∈ N. Hn also satisfies conditions (H3.4) and (H3.5), ∀ n ∈ N. This follows easily from the continuity of h and from the hypotheses (G3.4) and (G3.5). So, by Theorem 2.2, there exists a solution x for problem (H). Since h is a selection for the set valued map F (see (1) and (F 1.2)), the function x is also a solution of problem (P). By Theorem 3.1 we can easily deduce the following Corollary 3.2. Let Ω ⊂ RN , N ≥ 1, be a canonical domain such that int Ω 6= ∅ and let F : [0, T ] × Ω → P(RN ) be a set valued map such that the following conditions hold (F 1) ∃ θ : [0, T ] × Ω → P(RN ) such that (F 1.1) θ is l.s.c. in [0, T ] × Ω; (F 1.2) co θ(t, x) ⊆ F (t, x), ∀ (t, x) ∈ [0, T ] × Ω; (F 1.4) co θ(t, x) ∩ int TΩ (x) 6= ∅, ∀ (t, x) ∈ [0, T ] × Ω. Moreover, let G : [0, T ] × Ω → P(RN ) be a set valued map satisfying hypotheses (G1), (G2) and (G3) and let Ik : Ω → RN , k ∈ {1, . . . , p}, be an impulse function verifying conditions (I4) and (I5). Then, there exists a solution of the impulsive periodic viable problem (P). Proof: Let us consider the set valued map ψ : [0, T ] × Ω → P(RN ) defined as follows ψ(t, x) = co θ(t, x) ∩ TΩ (x), ∀ (t, x) ∈ [0, T ] × Ω.
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It is enough to prove that ψ verifies hypotheses (F 1.1), (F 1.2) and (F 1.3) of Theorem 3.1. As TΩ (x) is convex and closed for any x ∈ Ω (see preliminaries), ψ has convex and closed values. Taking into account that co θ(t, x) ⊆ F (t, x), ∀ (t, x) ∈ [0, T ] × Ω, and the definition of ψ, we can say that ψ verifies hypotheses (F 1.2) and (F 1.3). Because θ satisfies property (F 1.4) and the set Ω is nonempty, closed and convex, we have that ψ is l.s.c. in [0, T ] × Ω (see Propositions 2.42, 5.35 and 2.54 of [14]). Remark 3.3. Let us observe that Theorem 3.1 strictly contains Corollary 3.2. Indeed, we can consider the set valued map F (t, x) = {0}, ∀ (t, x) ∈ [0, 1] × [0, 1].
4
Applications
In this section we are interested in the following impulsive periodic viable problem 0 x (t) ∈ F (t, x(t)) a.e. t ∈ [0, T ] \ {t1 , . . . , tp } (F) x(t+ k ) = x(tk ) + Ik (x(tk )) for any k ∈ {1, . . . , p} x(0) = x(T ),
where Ω ⊂ RN , N ≥ 1, is a canonical domain, F : [0, T ] × Ω → P(RN ) is a set valued map, 0 = t0 < t1 < · · · < tp+1 = T , Ik : Ω → RN is an impulse function for k ∈ {1, . . . , p}, and x(t+ ) = lim+ x(s). s→t
Remark 4.1. Let us note that by Theorem 3.1 and by Corollary 3.2 we can deduce two existence theorems for the impulsive periodic viable problem (F), by taking G(t, x) = {0}, ∀ (t, x) ∈ [0, T ] × Ω. If the set valued map F is l.s.c., from Theorem 3.1 and Corollary 3.2 we can deduce the following results about the existence of periodic solutions for the impulsive viable problem (F): Corollary 4.2. Let Ω ⊂ RN , N ≥ 1, be a canonical domain such that int Ω 6= ∅ and let F : [0, T ] × Ω → P(RN ) be a set valued map such that the following conditions hold (F 2) F (t, x) is nonempty, convex, closed, ∀ (t, x) ∈ [0, T ] × Ω; (F 3) F is l.s.c. in [0, T ] × Ω; (F 4) F (t, x) ⊆ TΩ (x), ∀ (t, x) ∈ [0, T ] × Ω; and let Ik : Ω → RN , k ∈ {1, . . . , p}, be an impulse function satisfying hypotheses (I4) and (I5). Then, there exists a solution of the impulsive periodic viable problem (F).
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Corollary 4.3. Let Ω ⊂ RN , N ≥ 1, be a canonical domain such that int Ω 6= ∅ and let F : [0, T ] × Ω → P(RN ) be a set valued map such that the following conditions hold (F 2) F (t, x) is nonempty, convex, closed, ∀ (t, x) ∈ [0, T ] × Ω; (F 3) F is l.s.c. in [0, T ] × Ω; (F 5) F (t, x) ∩ int TΩ (x) 6= ∅, ∀ (t, x) ∈ [0, T ] × Ω; and let Ik : Ω → RN , k ∈ {1, . . . , p}, be an impulse function satisfying hypotheses (I4) and (I5). Then, there exists a solution of the impulsive periodic viable problem (F). Remark 4.4. We remark that none of the Corollaries 4.2 and 4.3 cover each other. Indeed, there exist set valued maps verifying hypotheses of Corollary 4.2, but not the conditions of Corollary 4.3. For example, we can consider the set valued map F : [0, 1] × [0, 1] → P(R) defined as follows F (t, x) = {0}, ∀ (t, x) ∈ [0, 1] × [0, 1]. Moreover, there exist set valued maps verifying hypotheses of Corollary 4.3, but not the conditions of Corollary 4.2. For example, let us consider the set valued map F : [0, 1] × [0, 1] → P(R) defined in this way F (t, x) = [−1, 1], ∀ (t, x) ∈ [0, 1] × [0, 1]. Remark 4.5. Let us note that, if F is a continuous single valued map, then Corollary 4.2 contains Corollary 4.3. Remark 4.6. We observe that Corollary 4.2 and Theorems 3.1 and 3.2 of [10] do not cover each other. Indeed there exist set valued maps verifying the hypotheses of Corollary 4.2, but not the conditions of Theorem 3.2 of [10] (and then of Theorem 3.1 of [10]). For example, we can consider the set valued map F : [0, 1] × [0, 1] → P(R) defined in this way {0} if t ∈ [0, 1], x = 1 F (t, x) = [0, 1] if t ∈ [0, 1], x ∈ [0, 1[.
Moreover, there exist set valued maps verifying the hypotheses of Theorem 3.1 of [10] (and then of Theorem 3.2 of [10]), but not the conditions of Corollary 4.2. For example, we can consider the second set valued map defined as in Remark 4.4. Finally, we note that Corollary 4.3 and Theorems 3.1 and 3.2 of [10] do not cover each other. Indeed, the set valued map F : [0, 1] × [0, 1] → P(R) defined as follows {−1} if t ∈ [0, 1], x = 1 F (t, x) = [−1, 1] if t ∈ [0, 1], x ∈ [0, 1[
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verifies the conditions of Corollary 4.3, but not the hypotheses of Theorem 3.2 of [10] (and then of Theorem 3.1 of [10]). Moreover, the first set valued map F defined as in Remark 4.4 satisfies Theorem 3.1 of [10] (and then Theorem 3.2 of [10]), but not the conditions of Corollary 4.3. In the field of single valued maps from Theorem 3.1 we obtain the following existence result. Theorem 4.7. Let Ω ⊂ RN , N ≥ 1, be a canonical domain such that int Ω 6= ∅ and let f : [0, T ] × Ω → RN be a single valued map such that the following conditions hold (f 3) ∃(fn )n∈N , fn : [0, T ] × Ω → RN such that (f 3.2) fn is continuous in [0, T ] × Ω, ∀ n ∈ N; (f 3.3) fn (t, x) ∈ TΩ (x), ∀ (t, x) ∈ [0, T ] × Ω, ∀ n ∈ N; (f 3.4) ∃ γ ∈ L∞ ([0, T ]) such that k fn (t, x) k≤ γ(t) a.e. on [0, T ], ∀ x ∈ Ω, ∀ n ∈ N; (f 3.5) a.e. t ∈ [0, T ], fn (t, ·) → f (t, ·) uniformly on Ω as n → ∞; and let Ik : Ω → RN , k ∈ {1, . . . , p}, be an impulse function satisfying hypotheses (I4) and (I5). Then there exists a solution of the impulsive periodic viable problem 0 x (t) = f (t, x(t)) a.e. t ∈ [0, T ] \ {t1 , . . . , tp } (Pf ) x(t+ k ) = x(tk ) + Ik (x(tk )) for any k ∈ {1, . . . , p} x(0) = x(T ).
Remark 4.8. Let us remark that hypotheses (f 3.2) and (f 3.5) imply only that f (t, ·) is continuous in Ω a.e. on [0, T ]. If the single valued map f is continuous in [0, T ] × Ω and verifies the property f (t, x) ∈ TΩ (x), ∀ (t, x) ∈ [0, T ] × Ω , then hypothesis (f 3) is trivially satisfied. Remark 4.9. Theorem 4.7 improves a result due to Hristova - Bainov (see Theorem 2 of [13]). Indeed, in Theorem 4.7 f is not necessarily continuous on [0, T ] × Ω (see Remark 4.8) and we do not require that f is a Lipschitz function with respect to the second variable. Moreover, hypothesis (f 3.3) weakens condition 3) in Theorem 2 of [13], being n
y ∈ RN : h ∇Φi (x), y i < 0, ∀ i ∈ α(x)
∀ x ∈ ∂Ω, ∀ t ∈ [0, T ].
o
= int TΩ (x),
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References [1] D.D. Bainov, V. Covachev: Impulsive differential equations with a small parameter, World Scientific, Series on Advances in Math. for Applied Sciences, 1994. [2]
D.D. Bainov, P.S. Simeonov: Systems with impulsive effect. Stability, theory and applications, Ellis Horwood Series in Maths and Appl., Ellis Horwood, Chicester, 1989. [3] D.D. Bainov, P.S. Simeonov: Impulsive differential equations. Asymptotic properties of the solutions, World Scientific, Series on Advances in Math. for Applied Sciences, 1995. [4] M.Benchohra, A.Boucherif: “Initial value problems for impulsive differential inclusions of first order”, Diff. Eqns. Dyn. Syst., Vol. 8, (2000), pp. 51–66. [5] M. Benchohra, A. Boucherif: “An existence result for first order initial value problems for impulsive differential inclusions in Banach spaces”, Arch. Math., Vol. 36, (2000), pp. 159–169. [6]
M. Benchohra, A. Boucherif, J.J. Nieto: “On initial value problems for a class of first order impulsive differential inclusions”, Disc. Math. Diff. Incl. Control Optim., Vol. 21, (2001), pp. 159–171.
[7] M. Benchohra, J. Henderson, S.K. Ntouyas: “On a periodic boundary value problem for first order impulsive differential inclusions”, Dyn. Syst. Appl., Vol. 10, (2001), pp. 477–488. [8] M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahabi: “Existence results for impulsive lower semicontinuous differential inclusions”, Int. J. Pure Appl. Math., Vol. 1, (2002), pp. 431–443. [9]
M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahabi: “Existence results for impulsive functional and neutral functional differential inclusions with lower semicontinuous right hand side”, Electron. J. Math. Phys. Sci., Vol. 1, (2002), pp.72– 91. [10] T. Cardinali, R. Servadei: “Periodic solutions of nonlinear impulsive differential inclusions with constraints”, Proc. AMS, Vol. 132, (2004), pp. 2339–2349. [11] B.C. Dhage, A. Boucherif, S.K. Ntouyas: “On periodic boundary value problems of first-order perturbed impulsive differential inclusions”, Electron. J. Diff. Eqns., Vol. 84, (2004), pp. 1–9. [12] M. Frigon, D. O’Regan: “Existence results for first order impulsive differential equations”, J. Math. Anal. Appl., Vol. 193, (1995), pp. 96–113. [13] S.G. Hristova, D.D. Bainov: “Existence of periodic solutions of nonlinear systems of differential equations with impulse effect”, J. Math. Anal. Appl., Vol. 125, (1987), pp. 192–202. [14] S. Hu, N.S. Papageorgiou: Handbook of multivalued analysis, Kluwer, Dordrecht, The Netherlands, 1997.
[15] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov: Theory of impulsive differential equations, World Scientific, Singapore, 1989. [16] V.D. Mil’man, A.D.Myshkis: “On the stability of motion in the presence of impulses”, Siberian Math. J., Vol. 1, (1960), pp. 233–237. [17] A.M. Samoilenko, N.A. Perestyuk: “Differential equations with impulse effect”, Visca Skola, Kiev, 1987. [in Russian]
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[18] P.J. Watson: “Impulsive differential inclusions”, Nonlin. World, Vol. 4, (1997), pp. 395–402.
CEJM 2(4) 2004 584–592
Folding theory applied to BL-algebras Young Bae Jun1∗ , Jung Mi Ko2† 1
Department of Mathematics Education Gyeongsang National University, Chinju 660-701, Korea 2 Department of Mathematics, Kangnung National University Gangneung, Gangwondo 210-702, Korea
Received 20 October 2003; accepted 21 June 2004 Abstract: The notion of n-fold grisly deductive systems is introduced. Some conditions for a deductive system to be an n-fold grisly deductive system are provided. Extension property for n-fold grisly deductive system is established. c Central European Science Journals. All rights reserved. ° Keywords: BL-algebra, (grisly) deductive system, n-fold grisly deductive system MSC (2000): 03G10, 03B52, 06B05
1
Introduction
Various problems in system identification involve characteristics which are essentially non-probabilistic in nature [11]. In response to this situation L. A. Zadeh introduced in 1965 fuzzy set theory as an alternative to probability theory. His fundamental idea consists in understanding lattice-valued maps as generalized characteristic functions of some new kind of subsets, so-called fuzzy sets, of a given universe. For historical reasons we quote the original definition (cf. [3] and [12]). Fuzzy logic grows as a new discipline from the necessity to deal with vague data and imprecise information caused by the indistinguishability of objects in certain experimental environments. As a set of mathematical tools, fuzzy logic is only using [0, 1]-valued maps and certain binary operations ∗ on the real unit interval [0, 1] known also as left-continuous t-norms. It took sometime to understand partially ordered monoids of the form ([0, 1], ≤, ∗) as algebras for [0, 1]-valued ∗ †
E-mail: [email protected] E-mail: [email protected]
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interpretations of a certain type of non-classical logic–the so-called monoidal logic. BLalgebras arise naturally in the analysis of the proof theory of propositional fuzzy logics. Indeed, Basic fuzzy logic, BL for short, and its corresponding BL-algebras, were introduced by H´ajek (see [4] and the references given there) with the purpose of formalizing the many-valued semantics induced by the continuous t-norms on the real unit interval [0, 1]. As a first step, H´ajek showed that a propositional formula is provable in BL if and only if it is a tautology in any linearly ordered BL-algebra. In [7], Ko and Kim investigated some properties of BL-algebras, and they [8] also studied relationships between closure operators and BL-algebras. Jun and Ko gave characterizations of a deductive system and discussed how to generate a deductive system by a set (cf. [5]). Turunen [9] studied Boolean deductive systems and implicative deductive systems. He proved that a BL-algebra L has a proper Boolean deductive system if and only if L is bipartite. In [6], Jun and Ko introduced the notion of grisly deductive systems of a BL-algebra. They gave conditions for a deductive system to be a grisly deductive system, and established an extension property for a grisly deductive system. In this paper we introduce the notion of n-fold grisly deductive systems, and give some conditions for a deductive system to be an n-fold grisly deductive system. We construct an extension property for n-fold grisly deductive system. BL-algebras, MV-algebras, and lattice implication algebras are closely related. Thus, all results in this paper will contribute much to studying MV-algebras and lattice implication algebras.
2
Preliminaries
Definition 2.1. [4, 10] A BL-algebra is an algebra (L,∧,∨,¯,Ã,0, 1) of type (2, 2, 2, 2, 0, 0) that satisfies the following conditions: (A1) (L, ∧, ∨, 0, 1) is a bounded lattice, (A2) (L, ¯, 1) is a commutative monoid, (A3) ¯ and à form an adjoint pair, i.e., z ≤ x à y if and only if x ¯ z ≤ y for all x, y, z ∈ L, where ≤ is the lattice ordering on L, (A4) (∀x, y ∈ L) (x ∧ y = x ¯ (x à y)), (A5) (∀x, y ∈ L) ((x à y) ∨ (y à x) = 1). Proposition 2.2. [4, 10] In a BL-algebra (L, ≤, ∧, ∨, ¯, Ã, 0, 1), we have the following properties: (p1) (∀x ∈ L) (x = 1 à x), (p2) (∀x ∈ L) (1 = x à x), (p3) (∀x, y ∈ L) (x ¯ y ≤ x, y), (p4) (∀x, y ∈ L) (x ¯ y ≤ x ∧ y), (p5) (∀x, y ∈ L) (y ≤ x à y), (p6) (∀x, y ∈ L) (x ¯ y ≤ x à y), (p7) (∀x, y ∈ L) (x ≤ y ⇔ 1 = x à y), (p8) (∀x, y ∈ L) (x = y ⇔ 1 = x à y = y à x),
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(p9) (p10) (p11) (p12) (p13)
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(∀x, y ∈ L) (x ¯ (x à y) ≤ y), (∀x, y, z ∈ L) (x à (y à z) = y à (x à z)). (∀x, y, z ∈ L) ((x à y) à (x à z) ≤ x à (y rsaz)), (∀x, y, z ∈ L) (x à y ≤ (z à x) à (z à y)), (∀x, y, z ∈ L) (x ≤ y ⇒ z à x ≤ z à y, y à z ≤ x à z).
Folding Theory
In what follows, let L denote a BL-algebra unless otherwise specified. For every a 1 , a2 ,· · · , an ∈ L, we define an if n = 1, P (a1 , a2 , · · · , an−1 \an ) := a1 Ã P (a2 , a3 , · · · , an−1 \an ) if n > 1, and P (an , b\x) = P (a, a, · · · , a, b\x) in which a occurs n-times.
Definition 3.1. [2, 9, 10] A subset D of L is called a deductive system of L if it satisfies the following conditions: (ds1) 1 ∈ D, (ds2) (∀x, y ∈ L) (x ∈ D, P (x\y) ∈ D ⇒ y ∈ D). Lemma 3.2. [7] Let D be a nonempty subset of L. Then D is a deductive system of L if and only if it satisfies: (ds3) (∀a, b ∈ D) (a ¯ b ∈ D), (ds4) (∀a ∈ D)(∀b ∈ L) (a ≤ b ⇒ b ∈ D). Definition 3.3. [6] Let D be a nonempty subset of L . Then D is called a grisly deductive system of L if it satisfies (ds1) and (ds5) (∀x, y, z ∈ L) (P (x, y\z) ∈ D, P (x\y) ∈ D ⇒ P (x\z) ∈ D). Note that every grisly deductive system is a deductive system, but the converse may not be true (see [6]). Proposition 3.4. If D is a grisly deductive system of L, then (i) (∀x, y ∈ L)(∀n ∈ N)(P (xn+1 \y) ∈ D ⇔ P (xn \y) ∈ D). (ii) (∀x, y, z ∈ L)(∀n ∈ N)(P (xn+1 , y\z) ∈ D ⇔ P (xn , y\z) ∈ D). Proof. (i) The necessity is straightforward. Assume that P (xn \y) ∈ D. Since x ≤ 1 for all x ∈ L, it follows from (p13) that P (xn \y) = P (1\P (xn \y)) ≤ P (x\P (xn \y)) = P (xn+1 \y) so from (ds4), P (xn+1 \y) ∈ D.
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(ii) Using (i) we have P (xn+1 , y\z) = P (xn+1 \P (y\z)) ∈ D ⇔ P (xn \P (y\z)) = P (xn , y\z) ∈ D. This completes the proof. Remark 3.5. The results (i) and (ii) of Proposition 3.4 do not hold in a deductive system of L in general. For example, consider a BL-algebra L in Example 3.7(1) below. Note that {1} is a deductive system of L (not a grisly deductive system of L), P (d\0) = a ∈ / {1}, 2 2 but P (d \0) = 1 ∈ {1}; and P (d, c\b) = a ∈ / {1} but P (d , c\b) = 1 ∈ {1}. Definition 3.6. Let D be a nonempty subset of L. Then D is called a (k, m; n)-fold grisly deductive system of L if it satisfies (ds1) and (ds6) there exist positive integers k, m and n such that (∀x, y, z ∈ L) (P (xk , y\z) ∈ D, P (xm \y) ∈ D ⇒ P (xn \z) ∈ D). If k = m = n in (ds6), we say that D is an n-fold grisly deductive system of L. Note that a 1-fold grisly deductive system is a grisly deductive system. Example 3.7. (1) Let L = {0, a, b, c, d, 1} be a set with Hasse diagram and Cayley tables as follows: ¯
1 a
b
c
d 0
Ã
1 a
b
c
d
0
1
1 a
b
c
d 0
1
1 a
b
c
d
0
c r JJr a ½ r rb d½
a
a
b
b
d 0 0
a
1 1 a
c
c
d
b
b
b
b
0
0 0
b
1 1
1
c
c
c
0
c
c
d 0
c
d 0
c
1 a
b
1 a
b
d
d
0 0 d 0 0
d
1 1 a 1 1 a
0
0
0 0 0
0
1 1
1r
JJ r
0 0
1 1 1
1
For every x, y ∈ L, define x ∧ y = x ¯ P (x\y) and x ∨ y = P (P (x\y)\y) ¯ P (P (P (x\y)\y)\P (P (y\x)\x)). Then (L; ≤, ∧, ∨, ¯, Ã, 0, 1) is a BL-algebra (cf. [6]). It is easy to verify that D := {1, a, b} is a (2, 3; 4)-fold grisly deductive system of L.
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(2) Let L = {0, a, b, c, 1} be a set with Hasse diagram and Cayley tables as follows:
1r
a r JJr b £ c r £ J J£r
0
¯
1 a
b
c
0
Ã
1 a
b
c
0
1
1 a
b
c
0
1
1 a
b
c
0
a
a a 0
c
0
a
1 1
b
c
0
b
b
0
b
0 0
b
1 a 1
c
c
c
c
c
0
c
0
c
1 1
1
b
0
0
0 0 0 0
0
1 1 1 1 1
b
For every x, y ∈ L, define x ∧ y = x ¯ P (x\y) and x ∨ y = P (P (x\y)\y) ¯ P (P (P (x\y)\y)\P (P (y\x)\x)). Then (L; ≤, ∧, ∨, ¯, Ã, 0, 1) is a BL-algebra. Then D := {1, a, c} is an n-fold grisly deductive system of L for every positive integer n. Theorem 3.8. Every grisly deductive system is a (k, m; n)-fold grisly deductive system for all positive integers k, m and n. Proof. Let D be a grisly deductive system of L and let x, y, z ∈ L be such that P (xk , y\z) ∈ D and P (xm \y) ∈ D. Then P (x, y\z) ∈ D and P (x\y) ∈ D by Proposition 3.4. It follows from (ds5) that P (x\z) ∈ D so from Proposition 3.4, P (x n \z) ∈ D. Hence D is a (k, m; n)-fold grisly deductive system of L. Corollary 3.9. Every grisly deductive system is an n-fold grisly deductive system for all positive integer n. The converse of Corollary 3.9 is not true in general. In fact, in Example 3.7(1), {1} is an n(≥ 2)-fold grisly deductive system which is not a grisly deductive system of L. This shows that an n-fold grisly deductive system may not be an n − 1-fold grisly deductive system in general. Theorem 3.10. For positive integers k, m and n, every (k, m; n)-fold grisly deductive system is a deductive system. Proof. Let D be a (k, m; n)-fold grisly deductive system of L. Taking x = 1 in (ds6) and using (p1), we conclude that P (y\z) ∈ D and y ∈ D imply z ∈ D. Thus D is a deductive system of L. Corollary 3.11. For a positive integer n, every n-fold grisly deductive system is a deductive system.
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The converse of Theorem 3.11 may not be true, i.e., there exist a deductive system D of L and a positive integer k such that P (xk , y\z) ∈ D and P (xk \y) ∈ D, but P (xk \z) ∈ / D. For example, consider the BL-algebra L in Example 3.7(1). Then {1} is a deductive system of L, but it is not a 1-fold grisly deductive system of L because P (d1 , a\0) = 1 ∈ {1} and P (d1 \a) = 1 ∈ {1}, but P (d1 \0) = a ∈ / {1}. Lemma 3.12. For a positive integer n, an n-fold grisly deductive system D of L satisfies the following implication: (ds7) (∀x, y ∈ L) (∀k ∈ N) (P (xn+k \y) ∈ D ⇒ P (xn \y) ∈ D). Proof. Let D be an n-fold grisly deductive system of L and let x, y ∈ L. If P (xn+1 \y) ∈ D, then P (xn \y) ∈ D. In fact, since P (xn , x\y) = P (xn+1 \y) ∈ D and P (xn \x) = 1 ∈ D, we have P (xn \y) ∈ D. Now, if P (xn+2 \y) ∈ D, then P (xn , x\P (x\y)) = P (xn+2 \y) ∈ D and P (xn \x) = 1 ∈ D. Hence P (xn+1 \y) = P (xn \P (x\y)) ∈ D, and so P (xn \y) ∈ D. Continuing this process, we can obtain the desired result. Theorem 3.13. Let n be a positive integer. If D is an n-fold grisly deductive system of L, then it is an n + k-fold grisly deductive system of L for every positive integer k. Proof. Let x, y, z ∈ L be such that P (xn+k , y\z) ∈ D and P (xn+k \y) ∈ D. Then P (xn+k \P (y\z)) ∈ D, and so P (xn , y\z) = P (xn \P (y\z)) ∈ D and P (xn \y) ∈ D by Lemma 3.12. Since D is an n-fold grisly deductive system of L, it follows that P (xn \z) ∈ D. Since P (xn+k \z) ≥ P (xn \z) and since every n-fold grisly deductive system is a deductive system, we get P (xn+k \z) ∈ D by (ds4). Hence D is an n + k-fold grisly deductive system of L. Theorem 3.14. Let D be a deductive system of L that satisfies the condition (ds7). Then D is an n-fold grisly deductive system of L Proof. Let x, y, z ∈ L be such that P (xn , y\z) ∈ D and P (xn \y) ∈ D. Since P (xn , y\z) ≤ P (xn \P (P (xn \y), xn \z)) = P (P (xn \y)\P (x2n \z)), it follows from (ds4) that P (P (xn \y)\P (x2n \z)) ∈ D so from (ds2), P (x2n \z) ∈ D. Using (ds7), we have P (xn \z) ∈ D, and so D is an n-fold grisly deductive system of L. We provide conditions for a deductive system to be an n-fold grisly deductive system. Theorem 3.15. Let D be a deductive system of L. Then the following are equivalent: (i) D is an n-fold grisly deductive system of L. (ii) (∀x, y ∈ L) (P (xn+1 \y) ∈ D ⇒ P (xn \y) ∈ D). (iii) (∀x, y, z ∈ L) (P (xn , y\z) ∈ D ⇒ P (P (xn \y)\P (xn \z)) ∈ D).
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Proof. (i) ⇒ (ii) is by Lemma 3.12. (ii) ⇒ (iii) Suppose that (ii) holds and let x, y, z ∈ L be such that P (x n , y\z) ∈ D. Since P (xn , y\z) ≤ P (xn \P (P (xn \y)\P (xn \z))), it follows from (ds4) that P (xn+1 \P (xn−1 \P (P (xn \y)\z))) = P (xn \P (P (xn \y)\P (xn \z))) ∈ D. Using (ii) we get P (xn+1 \P (xn−2 \P (P (xn \y)\z))) = P (xn \P (xn−1 \P (P (xn \y)\z))) ∈ D and so P (xn+1 \P (xn−3 \P (P (xn \y)\z))) = P (xn \P (xn−2 \P (P (xn \y)\z))) ∈ D by using (ii) again. Repeating this process, we obtain P (P (xn \y)\P (xn \z)) = P (xn \P (P (xn \y)\z)) ∈ D. (iii) ⇒ (i) Assume that (iii) is true and let x, y, z ∈ L be such that P (x n , y\z) ∈ D and P (xn \y) ∈ D. It follows from (iii) that P (P (xn \y)\P (xn \z)) ∈ D so from (ds2), P (xn \z) ∈ D. Hence D is an n-fold grisly deductive system of L. The following is a characterization of an n-fold grisly deductive system. Theorem 3.16. Let D be a subset of L that satisfies (ds1). Then D is an n-fold grisly deductive system of L if and only if it satisfies the following implication: (ds8) (∀x ∈ D) (∀y, z ∈ L) (P (y n+1 , x\z) ∈ D ⇒ P (y n \z) ∈ D). Proof. Suppose that D is an n-fold grisly deductive system of L and let x ∈ D and y, z ∈ L be such that P (y n+1 , x\z) ∈ D. Then P (x\P (y n+1 \z)) ∈ D and since D is a deductive system of L, it follows from (ds2) that P (y n+1 \z) ∈ D. Thus P (y n \z) ∈ D by Theorem 3.15(ii). Conversely, assume that D satisfies the condition (ds8). Let x, y ∈ L be such that P (x\y) ∈ D and x ∈ D. Then P (1n+1 , x\y) = P (x\y) ∈ D, and so y = P (1n \y) ∈ D by (ds8). Therefore D is a deductive system of L. Now let x, y ∈ L be such that P (xn+1 \y) ∈ D. Then P (xn+1 , 1\y) = P (xn+1 \y) ∈ D, and thus P (xn \y) ∈ D by (ds1) and (ds8). Therefore D is an n-fold grisly deductive system of L by Theorem 3.15. This completes the proof.
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Theorem 3.17. (Extension property for an n-fold grisly deductive system) Let D and E be deductive systems of L such that D ⊂ E. If D is an n-fold grisly deductive system of L, then so is E. Proof. Let x, y ∈ L be such that P (xn+1 \y) ∈ E. Since P (xn+1 \P (P (xn+1 \y)\y)) = 1 ∈ D, we have, by Theorem 3.15(ii), P (P (xn+1 \y)\P (xn \y)) = P (xn \P (P (xn+1 \y)\y)) ∈ D ⊂ E. It follows from (ds2) that P (xn \y) ∈ E. Applying Theorem 3.15, we conclude that E is an n-fold grisly deductive system of L.
4
Concluding remarks
In this paper, we have considered the folding theory of a grisly deductive system which is a generalization of a grisly deductive system in BL-algebras. We have provided conditions for a deductive system to be an n-fold grisly deductive system. We have established an extension property for an n-fold grisly deductive system. Since BL-algebras, M V algebras and lattice implication algebras are closely related, we will use the results of this paper to study ideals and filters (or, deductive systems) of M V -algebras, lattice implication algebras and related algebraic systems. Some important issues for future research are trying to find a new kind of deductive system in BL-algebras, and considering the fuzzification of n-fold grisly deductive systems as well as a new kind of deductive systems in BL-algebras.
5
Acknowledgements
The authors are highly grateful to referees for their valuable comments and suggestions for improving the paper. The first author is an Executive Research Worker of Educational Research Institute in GSNU.
References [1] L. Biacino and G. Gerla: “An extension principle for closure operators”, J. Math. Anal. Appl., Vol. 198, (1996), pp. 1–24. [2] A. DiNola, G. Georgescu and L. Leustean: “Boolean products of BL-algebras”, J. Math. Anal. Appl., Vol. 251, (2000), pp. 106–131. [3] J.A. Goguen: “L-fuzzy sets”, J. Math. Anal. Appl., Vol. 18, (1967), pp. 145–174. [4] P. H´ajek: Metamathematices of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998.
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[5] Y.B. Jun and J.M. Ko: “Deductive systems of BL-algebras”, Bull. Korean Math. Soc., [submitted]. [6] Y.B. Jun and J.M. Ko: “Grisly deductive systems of BL-algebras”, Bull. Korean Math. Soc., [submitted]. [7] J.M. Ko and Y.C. Kim: “Some properties of BL-algebras”, J. Korea Fuzzy Logic and Intelligent Systems, Vol. 11(3), (2001), pp. 286–291. [8] J.M. Ko and Y.C. Kim: “Closure operators on BL-algebras”, Comm. Korean Math. Soc., Vol. 19(2), (2004), pp. 219–232.. [9] E. Turunen: “Boolean deductive systems of BL-algebras”, Arch. Math. Logic, Vol. 40, (2001), pp. 467–473. [10] E. Turunen: Mathematics behind fuzzy logic, Springer-Verlag Co., Heidelberg, 1999. [11] L.A. Zadeh: “From circuit theory to system theory”, Proc. Inst. Radio Eng., Vol. 50, (1962), pp. 856–865. [12] L.A. Zadeh: “Fuzzy sets”, Inform. and Control, Vol. 8, (1965), pp. 338–353.
CEJM 2(4) 2004 593–604
Generalization of Weierstrass Canonical Integrals Olga Veselovska∗ Department of Applied Mathematics, National University ”Lviv Polytechnica”, S. Bandera str.,12 Lviv 79013 Ukraine
Received 21 May 2004; accepted 19 July 2004 Abstract: In this paper we prove that a subharmonic function in Rm of finite λ-type can be represented (within some subharmonic function) as the sum of a generalized Weierstrass canonical integral and a function of finite λ-type which tends to zero uniformly on compacts of Rm . The known Brelot-Hadamard representation of subharmonic functions in R m of finite order can be obtained as a corollary from this result. Moreover, some properties of R-remainders of λ-admissible mass distributions are investigated. c Central European Science Journals. All rights reserved. ° Keywords: Subharmonic function, Weierstrass canonical representation, subharmonic function of finite λ-type MSC (2000): 31B05
1
integral,
Brelot-Hadamard
Introduction
Throughout this paper Rm is the m-dimensional Euclidean space, S m is the unit sphere in Rm centered at the origin and ωm is its surface area. Let ∆ denote the Laplace operator. If u is a subharmonic function in Rm , then ∆u is non-negative in the sense of generalized functions, and µu = dm1ωm ∆u is a positive measure, which is called the Riesz mass distribution associated with u (see, e.g. [1, pp. 55–58], [13, p.43]). Here d2 = 1 and dm = m − 2 for m > 2. For any integer q ≥ 0, define ¯ P ¯ k q | yζ | cos kϕ ¯ ¯ , m = 2, ln ¯1 − yζ ¯ + k k=1 Kq (y; ζ) = ´i h³ q ¯ ¯k P ¯y¯ ν ζ y 1 , m > 2, , |ζ| ¯ ζ ¯ Ck |y| − |y−ζ|1m−2 + |ζ|m−2 k=0
∗
Email: [email protected]
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where ϕ is the angle between the radius-vectors of points y, ζ ∈ R2 ; (·, ·) is the scalar product in Rm , m > 2, and Ckν are the Gegenbauer polynomials [2, pp. 302, 329], [12, p.125] of degree k and order ν = (m − 2)/2. Let µ be a mass distribution in Rm such that 0 6∈ suppµ and p = pµ denotes the least R∞ d µ(t) nonnegative integer number for which tm+p−1 < ∞. 0
The function
Jp (y; µ) =
Z
Kp (y; ζ)dµ(ζ)
|ζ|<∞
is called the Weierstrass canonical integral of genus p [1, p. 78], [13, pp. 67–68]. It is a subharmonic function and µ is its associated Riesz mass distribution (see, e.g. [3, p. 163]). Let u be a subharmonic function in Rm which is harmonic in some neighborhood of the origin, with u(O) = 0, and let λ be a positive continuous increasing function on (0, +∞), which is called the function of growth. Put B(r, u) = max {u(y) : |y| 6 r}. Definition 1.1. [4] A subharmonic function u is called a function of finite λ-type if there exist constants A and B such that B(r, u) 6 A λ(Br) for all r > 0. The class of such functions is denoted by Λs . It is known (see [5], [13, pp. 68–69]) that in the case λ(r) = r % , % > 0, the subharmonic function u ∈ Λs is represented in the form of sum u(y) = Jp (y; µu ) + Pn (y),
(1)
where p 6 % and Pn is a harmonic polynomial of degree n 6 %. If we denote Z uR (y) = Kp (y; ζ) dµu (ζ) + Pn (y) (R > 0) |ζ|6R
the sum (1) can be written as u(y) = uR (y) +
Z
Kp (y; ζ) dµu (ζ).
|ζ|>R
In addition, the Riesz mass distributions associated with the functions u and uR coincide in the ball {y ∈ Rm : |y| 6 R}, the function u − uR converges to zero uniformly on compacts of Rm as R → ∞. Moreover, each of functions u, uR , u − uR belongs to the class Λs (see, e.g. [1, pp. 79–80]). We shall generalize this result to the case of arbitrary functions u ∈ Λs , subharmonic in Rm , m > 3, with a more general growth. The analogous generalization for entire
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functions f in the plane, such that ln |f | ∈ Λs was obtained by L.Rubel [6] and for subharmonic in R2 functions of finite λ-type by Ya.Vasylkiv [7]. Let α[λ] denote a lower order of λ defined by ln λ(r) . r→∞ ln r
α[λ] = lim
Theorem 1.2. For every function u ∈ Λs there exist a subharmonic function h 6≡ −∞, an unbounded set < of positive numbers and a family {uR : R ∈ <} of subharmonic functions such that 1) the Riesz mass distributions associated with the functions uR and u + h coincide in the ball {y ∈ Rm : |y| 6 R} for all R ∈ <; 2) the difference (u+h)−uR tends to 0 uniformly on compacts of Rm as R → ∞, R ∈ <; 3) h, uR , (u + h) − uR ∈ Λs for all R ∈ <. If α[λ] = ∞, then we can take h ≡ 0 and if ln λ(r) is convex in ln r, then we can take h ≡ 0 and < = {R : R > R0 } for some R0 > 0. Definition 1.3. The family of functions {uR : R ∈ <} defined by the preceding theorem is called the generalized Weierstrass canonical integral of function u. In the next section we obtain some auxiliary results, which will be used for the proof of Theorem.
2
The remainders
Definition 2.1. [4] A mass distribution µ in Rm , 0 6∈ supp µ, is called λ-admissible, if there exist constants A, B and l ∈ R+ such that ¯ ¯ ¯ Z ¯ ·µ · ¸ ¶¸ ¯ ¯ y dµ(y) λ(Br2 ) ¯ ¯ ν l λ(Br1 ) Ck x, + ¯ ¯ 6 A(k + 1) ¯ |y| |y|k+2ν ¯ r1k r2k ¯ r1 <|y|6r2 ¯
(2)
for all r1 , r2 > 0, k ∈ Z+ , x ∈ S m . Here and below, ν = (m − 2)/2. Put V¯Rm = {y ∈ Rm : |y| 6 R}.
Definition 2.2. A mass distribution µR (R > 0) defined for any Borel set G ⊂ Rm by the equality µR (G) = µ(G \ V¯Rm ) is called the R-remainder of µ. Let < be a non-empty set of positive numbers. Definition 2.3. If the set < is unbounded, the family of remainders {µR : R ∈ <} is called complete.
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Definition 2.4. A family of remainders {µR : R ∈ <}, 0 6∈ supp µR , is called uniformly λ-admissible, if it satisfies the inequality (2) for all r1 , r2 > 0, k ∈ Z+ , x ∈ S m and R ∈ <. A spherical harmonic or a spherical Laplace function of degree k (k ∈ Z+ = {0, 1, 2, . . .}), denoted Y (k) , is defined as the restriction of a homogeneous, harmonic polynomial of degree k on the unit sphere S m (see, e.g. [8], [9]). Let µ be a mass distribution in Rm such that 0 6∈ supp µ, and let Y = {Y (k) (x)} (k ∈ Z+ , Y (0) (x) = 0) be some sequence of spherical harmonics. Definition 2.5. [4] The functions ck (x, r; Y, µ) = Y
(k)
k
(x) r + r
k
Z
Ckν
Ckν
·µ
|ζ|6r
−
1 rk+2ν
Z
|ζ|
k
|ζ|6r
·µ
ζ x, |ζ|
ζ x, |ζ|
¶¸
¶¸
dµ(ζ) |ζ|k+2ν
dµ(ζ) (k ∈ Z+ )
are called the spherical harmonics of the pair (Y, µ). Proposition 2.6. Let µ be a λ-admissible mass distribution in Rm . Then 1) there exists λ-admissible mass distribution µ0 > µ whose family of remainders {µ0R : R ∈ <} is complete and uniformly λ-admissible; (k) 2) for every such remainder µ0R there exists the sequence YR = {YR (x)} (k ∈ Z+ , (0) YR (x) = 0, R ∈ <) of spherical harmonics such that a) |ck (x, r; YR , µ0R )| 6 A(k+1)l λ(Br)
(3)
for all r > 0, k ∈ Z+ , x ∈ S m , R ∈ < and some positive constants A, B and l ∈ R+ ; b)
lim ck (x, r; YR , µ0R ) = 0
(4)
<3R→∞
for all r > 0, k ∈ Z+ , x ∈ S m . If α[λ] = ∞, then we can take µ0 = µ. If ln λ(r) is convex in ln r, then we can take µ0 = µ and {µ0R } = {µR : R > R0 > 0}. The following lemma from [6] will be used in the proof of the last special case of Proposition. Lemma 2.7. If ln λ(r) is convex in ln r, then there is R0 > 0 such that for every R > R0 we can find σ = σ(R) > 0, for which λ(Br) λ(BR) = inf r>0 Rσ rσ holds. Here B is some positive constant.
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Proof of statement 1) of Proposition. For r > 0, k ∈ Z+ , x ∈ S m and any mass distribution µ in Rm , we put ·µ ¶¸ Z dµ(y) y ν Ik (r; x, µ) = Ck x, , |y| |y|k+2ν |y|6r
Ik (r1 , r2 ; x, µ) = Ik (r2 ; x, µ) − Ik (r1 ; x, µ) (r1 6 r2 ). It follows from the equation C0ν (t) = 1 [2, p.176], [12, p.125] that the functions I0 (r; x, µ), I0 (r1 , r2 ; x, µ) are independent of x. We distinguish three cases. I. Let ln λ(r) be a convex in ln r function, let µ be a λ-admissible mass distribution in Rm and the numbers R, σ are as in Lemma 1. We shall show that inequality (2) holds for R-remainder of µ at R > R0 , where R0 is defined in Lemma 2.7. If r1 6 r2 6 R, then Ik (r1 , r2 ; x, µR ) = 0. If R 6 r1 6 r2 , then Ik (r1 , r2 ; x, µR ) = Ik (r1 , r2 ; x, µ) and therefore the mass distribution µR satisfies inequality (2) for all R ∈ <. If r1 6 R 6 r2 , then Ik (r1 , r2 ; x, µR ) = Ik (R, r2 ; x, µ). The last expression doesn’t exceed · ¸ λ(Br2 ) l λ(BR) A(k + 1) + , Rk r2k where A, B are some positive constants and l ∈ R+ . Let k > σ. Then, by Lemma 1, λ(BR) 1 λ(Br1 ) λ(BR) λ(Br1 ) 1 = · k−σ 6 . · k−σ = σ k σ R R R r1 r1k r1 Suppose now that k < σ. In this case we have λ(BR) λ(Br2 ) λ(BR) 1 λ(Br2 ) 1 = . = · 6 · Rk Rσ Rk−σ r2σ r2k r2k−σ Thus
hence
λ(BR) 6 max Rk
½
λ(Br1 ) λ(Br2 ) , r1k r2k
|Ik (r1 , r2 ; x, µR )| 6 2A(k + 1)
l
·
¾
λ(Br1 ) λ(Br2 ) + r1k r2k
¸
for all r1 , r2 > 0, k ∈ Z+ , x ∈ S m , R ∈ <. If we choose µ0 = µ whose complete family of remainders is {µR : R > R0 }, the statement 1) of Proposition is proved in the case I. II. Let α[λ] = ∞, λ(0) > 0. For any positive σ, put n o Rσ = max R : λ(BR)/Rσ = inf λ(Br)/r σ , r>0
where B is some positive constant. Since lim λ(Br)/r σ = ∞ for every σ > 0 and λ r→∞ is a continuous function, the numbers Rσ are defined correctly and they are positive.
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As it is shown in [6] Rσ is an increasing unbounded function of σ. Therefore the family of remainders {µRσ : σ > 0} is complete. Analogously to case I it is easy to verify that this family is uniformly λ-admissible. III. Let α[λ] < ∞. Then there exists d ∈ N∗ = {1, 2, 3, . . . } such that lim λ(r)/rb = 0 r→∞
for all b > d. We denote by < the set of positive numbers R satisfying the relation ½ ¾ λ(Br) λ(BR) . = inf r6R Rd rd It is obvious that
lim λ(BR)/Rd = 0. <3R→∞
Let construct the mass distribution µ0 in the following way. Consider the function d Γ(ν) X Q(η; ξ) = D − ν+1 (j + ν)Cjν [(η, ξ)], 2π j=1
(5)
where the constant D is chosen in such a way that Q(η, ξ) > 0 for all η ∈ S m and all ξ ∈ S m. Let µ be λ-admissible mass distribution in Rm and let ϕ be an arbitrary function in the class C0 (Rm ) of continuous functions in Rm with compact support. Put Z L(ϕ) = Ψ(ϕ; y)dµ(y), Rm
where Ψ(ϕ; y) =
Z
ϕ(tη) Q(η; ξ) dS(η) (y = tξ,
t = |y|,
ξ ∈ S m ).
Sm
It is easy to see that L(ϕ) is a linear continuous positive functional defined on C 0 (Rm ). We continue the functional L on the class of semicontinuous functions as this is e If χG is the done in [3, pp. 105–114] and denote the obtained continuation by L. characteristic function of the set G ⊂ Rm , we define the measure µ e associated with e e the functional L by µ e(G) = L(χG ). Let χ1 and χ2 be the characteristic functions of the balls V¯rm and V¯rm respectively. Since 1 2 0 6∈ supp µ e, then in some neighborhood of the point y = 0, which doesn’t intersect with supp µ e, we change the function Ckν [(x, y/|y|)]/|y|k+2ν (x is fixed), so that it becomes continuous in V¯rm and hence in V¯rm . Therefore we have 1 2
Denote
Ik (r1 , r2 ; x, µ e) = Ik (r2 ; x, µ e) − Ik (r1 ; x, µ e) h³ h³ ´i ´i y y Ckν x, |y| Ckν x, |y| e χ2 e χ1 −L . =L |y|k+2ν |y|k+2ν Fk (x; y) =
Ckν
´i h³ y x, |y|
|y|k+2ν
,
(6)
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Fk+ (x; y) = max{0; Fk },
599
Fk− (x; y) = − min{0; Fk }.
Then Fk = Fk+ − Fk− and e i Fk ) = L(χ e i F + ) − L(χ e i F − ). L(χ k k
(7)
Here and below index i takes values 1,2. Since the balls V¯rm are compact subsets in Rm , the functions χi are upper semiconi tinuous. Then, according to Theorem 1.4 from [3, p.22], there exist decreasing sequences {gni } of continuous functions such that gni → χi as n → ∞. It is obvious that every sequence {gni } can be chosen so that the supports of functions gni are contained in some compact. Further, since Fk+ is a nonnegative continuous function, the sequences {gni Fk+ } are monotone (decreasing in n) sequences of continuous functions with compact supports, moreover, gni Fk+ → χi Fk+ as n → ∞. By Theorem 3.3 from [3, p.109] and definition of the functional L, Z + + i e L(χi Fk ) = lim L(gn Fk ) = lim Ψ(gni Fk+ ; y) dµ(y). n→∞
n→∞ Rm
From the non-negativity of function Q we conclude that sequences {gni Fk+ Q} are monotone decreasing in n. Hence the sequences {Ψ(gni Fk+ ; y)} are also monotone decreasing in n. Using Lebesgue’s theorem about monotone convergence, we have Z + e L(χi Fk ) = Ψ(χi Fk+ ; y) dµ(y). Rm
e i F −) = Analogously L(χ k
R
Rm
Ψ(χi Fk− ; y) dµ(y).
Thus, taking into account equality (7), we obtain Z e i Fk ) = L(χ Ψ(χi Fk ; y) dµ(y). Rm
Hence, by means of (6), we find Z Z ν Ck [(x, η)] Ik (r1 , r2 ; x, µ e) = χ1,2 (|y|η) Q(η; ξ) dS(η) dµ(y) |y|k+2ν Sm
Rm
=
Z
r1 <|y|6r2
1
|y|k+2ν
Z
Sm
Ckν [(x, η)] Q(η; ξ) dS(η)
dµ(y),
(8)
where y = |y|ξ, ξ ∈ S m , and χ1,2 is a characteristic function of ring {y ∈ Rm : r1 < |y| 6 r2 }. Denote Z Ckν [(x, η)] Q(η; ξ) dS(η), Dk (x; ξ) = Sm
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where the function Q is defined by relation (5). At k = 0 in consequence of orthogonality of Gegenbauer polynomials [8, p. 179] we have D0 (x; ξ) = Dωm . In the case 0 < k 6 d, using the equalities [2, p. 238] Z 0, k 6= j, ν ν Ck [(x, η)] Cj [(η, ξ)] dS(η) = ν+1 ν 2π Ck [(x,ξ)] , k = j, m (k+ν)Γ(ν)
S
we get Dk (x; ξ) = −Ckν [(x, ξ)]. If k > d, then Dk (x; ξ) = 0. Therefore from (8) we conclude that D ωm I0 (r1 , r2 ; x, µ), k = 0, Ik (r1 , r2 ; x, µ e) = −Ik (r1 , r2 ; x, µ), 0 < k 6 d, 0, k > d.
Choose µ0 = µ + µ e. Then, by virtue of the previous relations, the equalities (1 + D ωm )I0 (r1 , r2 ; x, µ), k = 0, Ik (r1 , r2 ; x, µ0 ) = 0, 0 < k 6 d, Ik (r1 , r2 ; x, µ), k>d
hold. Therefore it remains to verify that the remainders µ0R (R ∈ <) are uniformly λadmissible when k = 0 and k > d. For this it is sufficient to consider the case r1 6 R 6 r2 . Let k = 0. Then |I0 (r1 , r2 ; µ0R )| = (1+D ωm )|I0 (R, r2 ; µ)| 6 (1+D ωm )|I0 (r1 , r2 ; µ)| 6 A[λ(Br1 )+λ(Br2 )], where A and B are some positive constants. If k > d, we have |Ik (r1 , r2 ; x, µ0R )|
= |Ik (R, r2 ; x, µ)| 6 A(k + 1)
l
·
λ(BR) λ(Br2 ) + Rk r2k
¸
for A, B > 0 and l ∈ R+ . But λ(Br1 ) λ(BR) λ(BR) 1 λ(Br1 ) 1 = · k−d 6 · k−d = d k d R R R r1 r1k r1 and the proof of statement 1) of Proposition is completed. Proof of statement 2) of Proposition. At first let prove relation (3). Let k = 0. For arbitrary mass distribution µ in Rm , define Zr n(t, µ) dt, N (r, µ) = (m − 2) tm−1 0
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where n(t, µ) =
R
601
dµ(τ ). Then the necessary relation can be obtained from the equa-
|τ |≤t tion c0 (x, r; YR , µ0R ) = N (r, µ0R ) remainders {µ0R : R ∈ <}. ∗
(see [4]) and the uniform λ-admissibility of family of
n o (k) Suppose now that k ∈ N . In this case we shall choose the sequence YR (x) of spherical harmonics in the same way as in [10] and [4]. If lim λ(Br)r −k > 0 holds for all k, we put p[λ] = ∞, otherwise we put r→∞
¾ ½ −k p[λ] = min k : lim λ(Br)r = 0 . r→∞
© ª Let 1 6 k < p[λ]. Then inf λ(Br)r −k : r > 0 > 0. Therefore for such k there exists (k) rk such that λ(Brk )rk−k 6 2λ(Br)r −k for all r > 0. In this case we choose YR (x) = −Ik (rk ; x, µ0R ). Suppose that k > p[λ]. Then there is the sequence {%j }, %j ↑ ∞ as j → ∞ such that −p[λ]
lim λ(B%j ) %j
j→∞
= 0.
Since |Ik (%i ; x, µ0R )
−
Ik (%j ; x, µ0R )|
6 A(k + 1)
l
"
(9)
λ(B%i ) λ(B%j ) + %ki %kj
#
for A, B > 0 and l ∈ R+ , then from (9) we find that the sequence {Ik (%j ; x, µ0R )}j∈N∗ (k)
is a Cauchy sequence for fixed x, k and R. Therefore for k > n p[λ] we o put YR (x) = (k) 0 − lim Ik (%j ; x, µR ). By virtue of such choice of sequence YR = YR (x) , we have j→∞
¯ ¯ ¯ ¯ ·µ ¶¸ Z ¯ ¯ 0 y dµ (y) ¯ (k) ¯ R ν Y (x) + C x, ¯ R ¯ = |Ik (r; x, µ0R ) − Ik (rk ; x, µ0R )| k k+2ν ¯ ¯ |y| |y| ¯ ¯ |y|6r 6 A(k + 1)
l
·
¸ λ(Brk ) λ(Br) λ(Br) + 6 3A(k + 1)l k k r rk rk
for 1 6 k < p[λ] and ¯ ¯ ¯ ¯ ·µ ¶¸ Z ¯ ¯ 0 y dµ (y) ¯ ¯ (k) R Ckν x, ¯ = lim |Ik (r; x, µ0R ) − Ik (%j ; x, µ0R )| ¯YR (x) + k+2ν ¯ j→∞ ¯ |y| |y| ¯ ¯ |y|6r 6 A(k + 1)l
"
# λ(Br) λ(B%j ) l λ(Br) + lim 6 A(k + 1) j→∞ rk rk %kj
for k > p[λ]. Therefore ¯ ¯ ¯ Z µ ¶ ¯ ¶¸ ·µ k ¯ ¯ |y| y 1 ¯ ¯ dµ0R (y)¯ . Ckν x, |ck (x, r; YR , µ0R )| 6 3A(k + 1)l λ(Br) + 2ν ¯ ¯ ¯ r r |y| ¯|y|6r ¯
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Applying inequality n(r, µ0 )/(2r2ν ) 6 N (2r, µ0 ) (see [4]) to the last addend, we find ¯ ¯ ¯ Z µ ¶ ¯ ·µ ¶¸ Z k ¯ 1 ¯¯ y |y| 1 ν ¯ ν 0 dµ0R (y) Ck x, dµR (y)¯ 6 2ν Ck (1) ¯ 2ν ¯ ¯ r r |y| r ¯|y|6r ¯ |y|6r
n(r, µ0R ) 6 2 Ckν (1) N (2r, µ0R ). 2ν r ν 2ν−1 Hence, taking into account relation Ck (1) = O (k ), k → ∞ (see [9]) and uniform 0 λ-admissibility of family of remainders {µR : R ∈ <}, we obtain estimate (3) for some possibly other constants A, B and l = m − 3. It remains to proof the relation (4). Since the integrals ¶¸ 0 ¶¸ ·µ ·µ Z Z dµR (y) y y k ν ν , |y| Ck dµ0R (y) Ck x, x, |y| |y|k+2ν |y| = Ckν (1)
|y|6r
|y|6r
(k)
are equal to zero for all R > r, then it is sufficient to show that YR (x) → 0 as < 3 R → (k) ∞. The last is obvious from the definition of spherical harmonics YR possibly with the exception of ¯ ¯ the case k > p[λ], p[λ] < ∞. In this case by the inequation (3), we have ¯ (k) k¯ ¯YR (x) r ¯ 6 A(k + 1)l λ(Br) for r < R and therefore (lettinq r → R) ¯ ¯ λ(BR) ¯ (k) ¯ l λ(BR) 6 A(k + 1) ¯YR (x)¯ 6 A(k + 1)l Rk Rp[λ]
since k > p[λ]. From the construction of family < (with d = p[λ] in section III of the proof of state(k) ment 1) of Proposition) it follows that lim λ(BR)/Rp[λ] = 0, from which lim YR (x) = <3R→∞
<3R→∞
0. This completes the proof of Proposition. ∞ P Let f ∈ L1 (S m ), then the series Y (k) (x; f ) is called its Fourier-Laplace series. Here k=0
(k)
(k)
(k)
(k)
(k) Y (k) (x; f ) = a1 Y1 (x) + a2 Y2 (x) + . . . + a(k) γk Yγk (x),
(k)
(k)
(k)
(k)
(k)
{Y1 , Y2 , . . . , Yγk } is the orthonormal base, ai = (f, Yi ) In the case m = 2 we have the trigonometric Fourier series. Denote ur (x) = u(rx), r > 0, x ∈ S m .
(i = 1, 2, . . . , γk ).
Definition 2.8. [10] The functions ck (x, r; u) = Y (k) (x; ur ) (k ∈ Z+ , x ∈ S m ) are called the spherical harmonics associated with the function u. Lemma 2.9. Let < be an unbounded set of positive numbers, and let {gR : R ∈ <} (gR (0) = 0) be a family of subharmonic functions such that
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a) the Riesz mass distribution associated with the function gR is equal to 0 in the ball V¯Rm ; b) | ck (x, r; gR )| 6 A(k + 1)l λ(Br) for some positive constants A, B, l ∈ R+ and for all r > 0, x ∈ S m , k ∈ Z+ , R ∈ <; c) lim ck (x, r; gR ) = 0 <3R→∞
for all k ∈ Z+ , r > 0 and x ∈ S m . Then
lim gR (y) = 0 uniformly on compacts of Rm . <3R→∞
Proof. Since by virtue of condition a) the function gR is harmonic in the ball V¯Rm , we can apply Poisson-Jensen’s formula [3, pp. 139–140] to it. For r < r ∗ < R we obtain (r∗ )2ν gR (rx) = ωm
Z
[(r∗ )2 − r2 ] gR (r∗ ξ) dS(ξ) [(r∗ )2 − 2r∗ r(x, ξ) + r 2 ]ν+1
(x ∈ S m ),
Sm
where ( · , · ) denotes the scalar product in Rm . Expanding the Poisson integral in series in spherical harmonics (see [10]), we have gR (rx) =
∞ ³ X r ´k k=0
Choose r∗ = 2r. Then |gR (rx)| 6
∞ P
r∗
Y (k) (x; (gR )r∗ ).
2−k |ck (x, 2r; gR )|. For fixed r > 0 and x ∈ S m
k=0
the series
∞ P
2−k |ck (x, 2r; gR )| is functional one, defined on <. From condition b) by
k=0
Weierstrass indication this series converges uniformly on <. Let S(R) be its sum. Then lim S(R) = lim
R→∞
Therefore we get
R→∞
∞ X k=0
2−k |ck (x, 2r; gR )| =
∞ X k=0
2−k lim |ck (x, 2r; gR )| = 0. R→∞
lim gR (rx) = 0 uniformly in r 6 r0 < r∗ .
<3R→∞
3
Proof of Theorem
Let µ = µu and let µ0 , <, YR and ck (x, r; YR , µ0R ) be such as in Proposition. By Theorem 1 from [4], in consequence of λ-admissibility of mass distribution µ0 , there exists function u∗ ∈ Λs whose Riesz mass distribution is µ0 . We shall assume that u∗ (0) = 0. According to Lemma 4 from [4], there are subharmonic functions vR , with vR (0) = 0, such that ck (x, r; vR ) = ck (x, r; YR , µ0R ) for all r > 0, x ∈ S m , k ∈ Z+ and also µvR = µ0R for every R ∈ <. Since the mass distributions µ0R are λ-admissible, then by Theorem 1 from [4] functions vR belong to the class Λs . Moreover, by Lemma 2 when < 3 R → ∞ these functions tend to 0 uniformly on compacts of Rm . Put uR = u∗ − vR and h = u∗ − u. It is obvious that the functions uR and h are subharmonic and their Riesz mass distributions satisfy condition 1) of Theorem. Since
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(u + h) − uR = u∗ − uR = vR , condition 2) is true. Condition 3) we obtain from [11, Theorem 1]. The last conditions of Theorem immediately follow from Proposition. The well-known representation of subharmonic functions in Rm of finite order can be obtained as a corollary from Theorem, if we take λ(r) = r β , β > %, [β] = q (with % and q as in the introduction), h ≡ 0, < = {R : R > R0 } at some R0 > 0, and R uR (y) = K(y; ζ) dµ(ζ) + ΦR (y), where K(y; ζ) = −|y − ζ|−2ν , and ΦR (y) is harmonic |ζ|6R
in y for every R ∈ <.
References [1] L.I. Ronkin: Introduction into the theory of entire functions of several variables, Nauka, Moscow, 1971. [Russian] [2] A. Bateman, A. Erdelyi: Higher transcendental functions, 2, Nauka, 1974. [Russian] [3] W.K. Hayman, R.B. Kennedy: Subharmonic functions, Acad. Press, London, 1976. [4] A.A. Kondratyuk: “Spherical harmonics and subharmonic functions (Russian)”, Mat. Sb., Vol. 125, (1984), pp. 147–166. [English translation in Math. USSR, Sb. 53, (1986), pp. 147–167] [5] B. Azarin: Theory of growth of subharmonic functions, Texts of lecture, part 2, Krakov, (1982). [Russian] [6] vL.A. Rubel: A generalized canonical product, In trans.: Modern problems of theory of analytic functions, Nauka, Moscow (1966), pp. 264-270. [7] Ya.V.Vasylkiv, An investigation asymptotic characteristics of entire and subharmonic functions by method of Fourier series, Abstract dissertation, Donetsk, 1986. (Russian) [8] E. Stein, G. Weiss: Introduction to Fourier analysis of Euclidean spaces, Princeton University Press, Princeton, New Jersey, 1971. [9] H. Berens, P.L. Butzer, S. Pawelke: “Limitierungs verfahren von Reihen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten”, Publs. Res. Inst. Math. Sci., Vol. 4, (1968), pp. 201–268. [10] A.A. Kondratyuk: “On the method of spherical harmonics for subharmonic functions (Russian)”, Mat.Sb., Vol. 116, (1981), pp. 147–165. [English translation in Math. USSR, Sb. 44, (1983), pp. 133-148] [11] O.V. Veselovska: “Analog of Miles theorem for δ-subharmonic functions in R m ”, Ukr. Math. J., Vol. 36, (1984), pp. 694–698. [Ukrainian] [12] N.N. Lebedev: Special functions and their applications, Revised edition, translated from the Russian and edited by Richard A. Silverman, Dover Publications, Inc., New York, 1972. [13] L.I. Ronkin: Functions of completely regular growth, translated from the Russian by A. Ronkin and I. Yedvabnik, Mathematics and its Applications (Soviet Series), Vol. 81, Kluwer Academic Publishers Group, Dordrecht, 1992.
CEJM 2(4) 2004 605–613
Approximation properties of wavelets and relations among scaling moments II V´aclav Finˇek∗ Institut f¨ ur Numerische Mathematik, Fakult¨at Mathematik und Naturwissenschaften, Technische Universitt Dresden, Zellescher Weg 12-14, 01069 Dresden
Received 11 March 2004; accepted 3 September 2004 Abstract: A new orthonormality condition for scaling functions is derived. This condition shows a close connection between orthonormality and relations among discrete scaling moments. This new condition in connection with certain approximation properties of scaling functions enables to prove new relations among discrete scaling moments and consequently the same relations for continuous scaling moments. c Central European Science Journals. All rights reserved. ° Keywords: Orthonormality, wavelets, approximation properties, scaling moments MSC (2000): 65T60, 42C40
1
Introduction
In many applications concerning wavelets, scalar products of arbitrary functions f with the scaling function have to be calculated. For deriving effective one-point quadrature formulas, the relation between the first scaling moment and the second one is crucial. The evaluations of scalar products was studied, e.g. in [7], where it was shown that the relation M2 = M12 is valid for the scaling function with a compact support which reproduces polynomials up to degree 2. Consequently, this relation was an essential tool used to derive one point quadrature formulas. Afterwards in [4] there were derived more relations among continuous scaling moments and the assumption on the compact support was removed. Strictly speaking for the orthonormal scaling function which reproduce exactly polynomials up to degree k, the validity of the following relations was proved: ∗
Email: [email protected]; [email protected]
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V. Finˇek / Central European Journal of Mathematics 2(4) 2004 605–613
n µ ¶ X n j=0
j
(−1)j+1 Mj Mn−j = 0
for n = 1, . . . , k − 1.
In the present paper, a new condition ensuring the orthogonality of the translations of the scaling function is derived. This condition shows that there is a close connection between the orthonormality and relations among discrete scaling moments. Furthermore, this new condition in connection with some approximation properties – scaling function reproduces exactly polynomials up to degree k - yield new relations among the discrete scaling moments: n µ ¶ X n j=0
j
(−1)j+1 mj mn−j = 0
for n = 1, . . . , 2k − 1.
(1)
At last the following is proved: if the relations among the discrete scaling moments (1) hold for n = 1, . . . , 2k − 1 then the same relations hold also among continuous scaling moments for n = 1, . . . , 2k − 1. This paper is organized as follows. Firstly, some basic approximation properties of scaling functions are summarized. Then the orthonormality conditions are discussed and one additional condition is derived. The last section deals with the relations among scaling moments. Initially, an auxiliary Lemma is showed. This Lemma describes connections between discrete and continuous scaling moments. Finally, in the last part of this section the relations among discrete scaling moments are proved and consequently the same relations for the continuous scaling moments.
2
Approximation properties
This part deals with the important question on scaling functions and wavelets - their accuracy of approximation. Besides, there is also a review of some important properties of scaling functions. As usual φ denotes the scaling function and hk the corresponding scaling coefficients. They are connected by the scaling equation: X φ(x) = 2 hk φ(2x − k). (2) k∈Z
Furthermore, ψ denotes the wavelet, gk the corresponding wavelets coefficients and X ψ(x) = 2 gk φ(2x − k) (3) k∈Z
is the wavelet equation (identity). Besides, it is assumed that translations and dilations of ψ form an orthonormal base of space L2 (R), and translations of φ form an orthonormal set – V0 denotes the space spanned by this set. Furthermore, the two fundamental operators in wavelet theory will be needed to characterize equivalent conditions for approximation properties and orthogonality: X M := {h2k−l }k,l∈Z and H(ω) := hk e−ikω . k∈Z
V. Finˇek / Central European Journal of Mathematics 2(4) 2004 605–613
607
Example 2.1. The matrix M for the scaling function corresponding with this scaling P equation φ(x) = 2 3k=0 hk φ(2x − k) has this form: ... h1 h0 0 0 0 0 0 0 0 0 h h h h 0 0 0 0 0 3 2 1 0 M = 0 0 0 h 3 h2 h1 h0 0 0 0 . 0 0 0 0 0 h 3 h2 h1 h0 0 . 0 0 0 0 0 0 0 h 3 h2 . . In wavelets applications, a function f is projected onto a scaling space Vj . The index j gives the scale 2−j in the representation and one basis for the space Vj is formed by the scaling function 2j/2 φ(2j x) and its translations by k 2−j . The projection fj is the part of f in that subspace, so it is a combination of those basis functions: fj (x) =
X
ajk 2j/2 φ(2j x − k)
∀j ∈ Z.
(4)
k∈Z
In contrast, wavelets come from splitting functions into several scales, i.e. multiresolution. This combines the small details at levels zero through j − 1 and the coarsest L L L one’s at V0 . For subspaces this is Vj = V0 W0 · · · Wj−1 . Except for V0 , the basis functions are now wavelets: fj (x) =
X k∈Z
a0k φ(x − k) +
j−1 X X
blk 2j/2 ψ(2l x − k)
∀j ≥ 0.
(5)
l=0 k∈Z
In practice, the level j is determined by balancing accuracy with cost. Both are approximately doubled when one level is added. The accuracy partly depends on the scaling coefficients and partly on the smoothness of the given function f . Here, a sufficient smoothness is assumed and our attention is paid to the scaling coefficients. A typical form of the error estimate in numerical analysis involves the n-th derivative of f : kf (x) − fj (x)k ≈ C 2−jn kf (n) (x)k.
(6)
The constant C and the exponent n depend on our choice of scaling coefficients. These determine φ and the subspaces. The key idea of this estimate is the following: Locally, every smooth function resembles a polynomial. The exponent n is the degree of the first polynomial that gives an error. The order of accuracy is determined by computing with polynomials. Mainly, our attention is paid to the conditions which ensure that combinations of φ(x − k) locally exactly reproduce the polynomials 1, x, . . . , xn−1 (also the order of accuracy is n). The test for accuracy of order n is Condition An which is recognized in the time domain, the frequency domain, and also in the eigenvalue domain:
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V. Finˇek / Central European Journal of Mathematics 2(4) 2004 605–613
Theorem 2.2. The order of accuracy is n if the scaling coefficients hk satisfy these three equivalent forms of Condition An : P k j for j = 0, 1, . . . , n − 1, 1. n sum conditions on the coefficients: k∈Z (−1) k hk = 0 1+e−iω n 2. H(ω) = ( 2 ) Q(ω) possess a zero of order n at π, 3. among eigenvalues of the matrix M are 1, 21 , . . . , ( 12 )n−1 . The three equivalent conditions are given and it remains to answer the question whether these conditions are sufficient for reproducing polynomials by combinations of scaling functions. The answer is provided by the following Theorem: Theorem 2.3. Let the values 1, 12 , . . . , ( 12 )n−1 be among eigenvalues of the matrix M and let yk denote left eigenvector of the matrix M related to the eigenvalue ( 12 )k . Then X
yk (l) φ(x + l) = xk
for k = 0, 1, . . . , n − 1,
(7)
l∈Z
where yk (l) is l-component of the vector yk . For proofs of the previous two Theorems and more details on this subject see [2, 3, 6]. Remark 2.4. The direct consequence of Theorem 2.3: Let the scaling coefficients hk satisfy the Condition An , then the wavelets orthogonal to {φ(x + l)}l∈Z are endowed with n vanishing continuous moments: Z
∞
xk ψ(x) dx = 0
for k = 0, 1, . . . , n − 1.
−∞
Remark 2.5. It is necessary to remark that the polynomials 1, x, . . . , xn−1 are not elements of the subspace V0 , despite being linear combinations of the translates of φ. Polynomials namely have infinite energy: Z
∞
(xk )2 dx = ∞. −∞
This contradicts the assumption V0 ⊂ L2 (R) based on the definition of V0 .
3
Conditions of orthonormality
This section brings together the requirements for an orthonormal scaling function. Recall that the 2π-periodic function H, denoted by: H(ω) :=
X
hk e−ikω ,
(8)
k∈Z
is the discrete Fourier transform of the sequence of hk and that these numbers are coordinates of φ(x/2). The function H is usually called ’quadrature mirror filter’ or ’conjugate
V. Finˇek / Central European Journal of Mathematics 2(4) 2004 605–613
609
quadrature filter’. Now, all concepts are introduced and some equivalent conditions of orthonormality can be recalled: P Theorem 3.1. Let φ(x) = 2 k∈Z hk φ(2x − k) then Z ∞ φ(x − m)φ(x)dx = δm,0 for m ∈ Z
(9)
−∞
implies: 1. |H(ω)|2 + |H(ω + π)|2 = 1, P 1 2. for m ∈ Z. j hj h2m+j = 2 δm,0
For the proof of this Theorem and more details on the subject of orthonormality see for instance [1, 3, 5, 6, 8]. Now, we derive a new equivalent condition which is an essential tool in the next section for deriving new relations among scaling moments. Theorem 3.2. Let φ(x) = 2
P2N −1 0
hk φ(2x − k) then
|H(ω)|2 + |H(ω + π)|2 = 1
(10)
is equivalent to:
where
¶ 2n µ X 1 2n δ0,n = (−1)i (ai a2n−i + bi b2n−i ) 2 i i=0 ai =
X
(2k)i h2k
and
bi =
for N > n ≥ 0, n ∈ N
X
(11)
(2k + 1)i h2k+1 .
Proof. For the purpose of this proof, the function H will be used in slightly different form: H(z) :=
2N −1 X
hk z k .
0
Then (10) is equivalent to the following equation: H(z)H(z −1 ) + H(−z)H(−z −1 ) = 1.
(12)
In order to simplify further this equation, H is decomposed into components H0 and H1 of H(z): H0 (z) =
N −1 X 0
h2k z
2k
and
H1 (z) =
N −1 X
h2k+1 z 2k+1 .
0
At this place it must be remarked that for orthonormal scaling functions, the number N is always odd. After replacing the operator H by the operators H0 and H1 , respectively (12) is equivalent to:
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1 (13) 2 and this is the well-known polyphase equation. Regarding (13) an operator X, defined by: H0 (z)H0 (z −1 ) + H1 (z)H1 (z −1 ) =
1 = 0, 2 includes only even polynomials in the form ck z 2k , where k runs from −N + 1 to N − 1 and at the same time ck = c−k for all k. Applying the operator (zD)n yields: X(z) := H0 (z)H0 (z −1 ) + H1 (z)H1 (z −1 ) −
(zD)n X(1) = 0
for n ≥ 0,
where D denotes derivative. However apparently only even derivatives up to degree N form the base for all functions in this form: cN −1 z 2N −2 + · · · + c1 z 2 + c0 z 0 + c1 z −2 + · · · + cN −1 z −2N +2 .
(14)
Because applying odd derivatives leads to the contradiction with the fact that the coefficients of z k and of z −k are the same. Furthermore, the system of functions in the form (14) contains only N free parameters and the even derivatives up to degree 2N − 2 are linearly independent and their number is also N . So, they form a basis of this system of functions. Thus (13) is equivalent to: (zD)2n X(1) = 0
for N − 1 ≥ n ≥ 0.
Finally, applying the operator (zD)2n on the operator X completes the proof: 1 δ0,n = (zD)2n (H0 (z)H0 (z −1 ) + H1 (z)H1 (z −1 ))(1) 2 ¶ 2n µ X 2n (−1)i (ai a2n−i + bi b2n−i ), = i i=0 where ai =
4
X
(2k)i h2k
and
bi =
X
(2k + 1)i h2k+1 .
Relations among scaling moments
In this part new relations among scaling moments are proved. First, let us use the following notations: Z X Mn := xn φ(x) dx and mn := hk k n k
V. Finˇek / Central European Journal of Mathematics 2(4) 2004 605–613
611
for the continuous and the discrete moments, respectively. Then the following relations among the continuous and the discrete moments hold: Lemma 4.1. For any n ∈ N the following relation holds true: n µ ¶ 1 X n mj Mn−j . Mn = n 2 j=0 j
(15)
Proof. By substitution, using the binomial formula, interchanging summations and interchanging summation and integration, respectively, the following is obtained: Z
n
x φ(x) dx = 2
Z X 1 x hk φ(2x − k) dx = n (y + k)n hk φ(y) dy 2 k k Z XX n µ ¶ n j n−j k y hk φ(y) dy j k j=0 n µ ¶Z n µ ¶ X X 1 X n n n−j j y φ(y) dy hk k = n mj Mn−j . 2 j=0 j j j=0 k
Z X
=
1 2n
=
1 2n
n
Now, the Theorem 3.2 will be used to derive new relations among the discrete moments and consequently this result will be an essential ingredient of the proof of the Theorem describing new relations among the continuous moments. P −1 Theorem 4.2. Let φ(x) = 2 2N hk φ(2x − k) and let φ be an orthonormal scaling 0 function with the order of accuracy m. Then the following formula holds: δ0,n
n µ ¶ X n (−1)i mi mn−i = i i=0
for n = 0, . . . , 2m − 1.
(16)
Proof. First, for odd number n the statement (16) always holds. So, it remains to prove this for even number n. The first condition from Theorem 2.2 implies: ai = b i
for 0 ≤ i < m
and in combination with the new derived condition of orthonormality (11) from Theorem 3.2 we have: δ0,n
n µ ¶ X n = (−1)i mi mn−i i i=0
for n = 0, . . . , 2m − 2.
At last it is necessary to remark the following: If the orthonormal scaling function has k scaling coefficients then firstly the number k must be even and secondly it is well-known that there are only k2 degrees of freedom for the filter coefficients. Thus the order of accuracy m can be at most N (so half of the number of the scaling coefficients). Hence
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the previous applying of Theorem 3.2 is correct, because the number of conditions of orthonormality from (11) is always greater or equal than the order of accuracy. P −1 Theorem 4.3. Let φ(x) = 2 2N hk φ(2x − k) and let φ be an orthonormal scaling 0 function with the order of accuracy m. Then the following formula holds: n µ ¶ X n j=0
j
(−1)j+1 Mj Mn−j = 0
for n = 1, . . . , 2m − 1.
(17)
Proof. By applying the Lemma 4.1, interchanging summations and using following relations among binomial coefficients: ¶ ¶ µ ¶µ µ ¶µ n n−i n n−k = k i i k
and
µ
n−i k
¶ ¶µ ¶µ ¶ µ n−i n−i−j k , = j k−j j
it yields: n X k=0
= = = = =
µ ¶ n (−1)k Mn−k Mk k µ ¶(n−k) X ¶ µ ¶k X n µ ¶ n−k µ k µ ¶ X n 1 n−k k 1 k (−1) mn−k−i Mi mk−j Mj k 2 2 i j i=0 j=0 k=0 µ ¶n X µ ¶µ ¶µ ¶ n X n−k X k 1 n−k k k n (−1) mn−k−i mk−j Mi Mj 2 k i j k=0 i=0 j=0 µ ¶µ ¶µ ¶ µ ¶n X n−i n−i X n X n−k k 1 k n (−1) mn−k−i mk−j Mi Mj k i j 2 i=0 j=0 k=j µ ¶n X µ ¶µ ¶µ ¶ n X n−i X n−i 1 n−i−j n−i k n (−1) mn−k−i mk−j Mi Mj 2 i k−j j i=0 j=0 k=j ! à n−i µ ¶ µ ¶µ ¶ µ ¶n X n−i n X X n − i − j n n − i 1 mn−k−i mk−j (−1)k−j Mi Mj (−1)j k − j i j 2 i=0 j=0 k=j
Applying the previous Theorem 4.2, the expression in the brackets acquires the value δj,n−i . Thus n µ ¶ X n k=0
and it results
µ ¶n X µ ¶µ ¶ n X n−i 1 n−i j n (−1) Mn−k Mk = (−1) Mi Mj δj,n−i k 2 i j i=0 j=0 µ ¶ µ ¶n X n 1 n−i n Mi Mn−i (−1) = i 2 i=0 k
V. Finˇek / Central European Journal of Mathematics 2(4) 2004 605–613
µ To conclude: n µ ¶ X n j=0
5
j
613
µ ¶n ¶ X n µ ¶ n 1 (−1)k Mn−k Mk = 0. 1− k 2 k=0
(−1)j+1 Mj Mn−j = 0
for n = 1, . . . , 2m − 1.
Acknowledgements
This work is partially supported by Deutsche Forschungsgemeinschaft (DFG) grant GR 1777/2-2, by N.HPRN-CT-2002-00286 (EU Project: Breaking Complexity), by Grant Agency of the Czech Republic - Grant No 201/04/1503 and by Ministry of Education of the Czech Republic - Grant No MSM 113200007.
References [1] A. Cohen, R.D. Ryan: “Wavelets and Multiscale Signal Processing (Transl. from the French)”, Applied Mathematics and Mathematical Computation, Vol. 11, (1995), pp. 232. [2] A. Cohen: “Wavelet methods in numerical analysis. Ciarlet”, P.G.(ed.) et al., Handbook of numerical analysis, Vol. 7 (Part 3); Techniques of scientific computing (Part 3), Elsevier, (2000), pp. 417-711. [3] I. Daubechies: “Ten Lectures on Wavelets”, CMBMS-NSF Regional Conference Series in Applied Mathematics, 61, Philadelphia, PA: SIAM, Society for Industial and Applied Mathematics, (1992), pp. 357. [4] V. Finˇek: “Approximation properties of wavelets and relations among scaling moments”, Numerical Functional Analysis and Optimization, (2002), [to appear] [5] A.K. Louis, P. Maass, A. Rieder: Wavelets - Theory and Applications Wiley, Chichester, 1997. [6] G. Strang, T. Nguyen: “Wavelets and Filter Banks - Gilbert Strang”, WellesleyCambridge Press, Vol. XXI, (1996), pp. 474. [7] W. Sweldens, R. Piessens: “Quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions”, SIAM J. Numer. Anal., Vol. 31, (1994), pp. 1240–1264. [8] P. Wojtaszczyk: “A Mathematical introduction to wavelets”, London Mathematical Society Student Text, Cambridge University Press, Vol. 37, (1997), pp. 261.
CEJM 2(4) 2004 614–614
Addendum to “Hochschild Cohomology of skew group rings and invariants” E. N. Marcos∗1 , R. Mart´ınez-Villa†2 , Ma. I. R. Martins‡1 1
Departamento de Matem´atica - IME, Universidade de S˜ao Paulo, C. Postal 66281, CEP 05315- 970, S˜ao Paulo, SP, Brasil 2 Instituto de Matematicas, UNAM- Campus Morelia, Apartado Postal 61-3, CP 58089, Morelia, Michoac´an, Mexico
The purpose of this addendum to [2] is twofold. First of all we want to remark that the proof of Theorem 2.9 proves in fact a more general result, which is the following. Theorem: ([2], Theorem 2.9) Let A be a k-algebra, G be a finite group acting on A and A[G] the associated skew group algebra. Then G acts on the Hochschild cohomology k-algebra HH • (A), and there is a ring monomorphism: HH • (A) ,→ HH • (A[G]). Consequently, we also can restate Proposition 3.5. Proposition: ([2], Proposition 3.5) Let G be a finite group and A be a G-graded ke be the covering algebra defined by the grading. Then G acts on HH • (A) e algebra. Let A • e • and there is a ring monomorphism from HH (A) into HH (A) Secondly, we want to observe that the statement in the last sentence of the introduction, related to Proposition 3.5, is not correct. This mistake came from a wrong interpretation of the results in [1].
References [1] C. Cibils and M. J. Redondo: “Cartan-Leray spectral sequence for Galois coverings of categories”, [to appear in Journal of Algebra]. [2] E. N. Marcos, R. Mart´ınez-Villa, Ma. I. R. Martins: “Hochschild cohomology of skew group rings and invariants”, Central European Journal of Mathematics, Vol. 2, (2004), pp. 177–191. ∗ † ‡
Email: [email protected] Email: [email protected] Email: [email protected]
